Lesson video

Hello there.

My name's Mr. Tilstone.

I hope you're ready for today's maths lesson, which is all about area.

We're going to be presenting you with some lovely, challenging, meaty problem solving opportunities to apply all your skills and knowledge all about area.

The answer to these questions won't come to you quickly.

It's going to take time, it's going to take thinking, it's going to take perseverance and a positive mindset.

If you're lucky enough to be able to work with somebody else, I would recommend doing that.

And when you solve these problems, it's going to feel like an amazing achievement.

So if you are ready, let's begin today's lesson.

The outcome of today's lesson is I can calculate the area of shapes made from two or more rectangles.

So we're going to do problem solving based on that.

And our keywords or keyword, we've only got one today.

My turn.

Compound.

Your turn.

Can you remember what compound means? Hopefully it's not an unfamiliar word too.

It's one you might have been using recently.

A compound shape is one that is made up of two or more shapes and they can be any shapes.

But we're going to focus specifically on rectangles.

So you can see, and the example on the screen, you've got a compound shape that's made of two rectangles.

Our lesson today is split into two cycles.

The first will be solving problems by decomposing rectangles.

And if you like geography and flags in particular, I think you're really going to enjoy cycle one.

And cycle two is efficient strategies with compound shapes.

But if you're ready, let's begin by solving problems by decomposing rectangles.

In this lesson, you're going to meet Lucas, Jun and Laura.

Have you met them before? They're going to be here today to give us a helping hand.

Jun's class are studying world flags in geography.

I wouldn't expect you to know this world flag, but it's the Indonesian flag.

It consists of two rectangles of equal size.

So the dimensions are the same, the areas' therefore the same.

And Jun measures the dimensions of this flag 12 centimetres by eight centimetres and that equals 96 centimetres square.

That's the area of the flag.

Do you think there's something that he could do with that number to work out the area of the red part or the area of the white part? Hmm.

The flag is a rectangle, but I could decompose it into two rectangles and calculate the area of each.

Yes you could.

And we're going to use division or (indistinct).

So 96 centimetres square divided by two equals 48 centimetres squared.

And that's the area of each colour.

So the red part is 48 centimetres squared and the white part is also 48 centimetres squared.

So we needed to do a little bit of thinking there.

We applied our skills, the skills that we knew and the knowledge that we knew about calculating the area of a rectangle and then we did something with it.

And that's going to be the theme of today's lesson.

It's a 48 centimetres squared.

But Lucas sees that problem a little bit differently.

He used a different strategy.

This is what he's done.

So he split one of the dimensions into four and four instead of eight.

So I treated it as two rectangles right from the start.

He didn't work out the area of the big rectangle.

He treated it as two from the start.

I calculated 12 centimetres by four centimetres twice.

I guess you've only gotta do that once, so it's going to be the same answer.

So 12 times four equals 48, therefore the area of each rectangle is 48 centimetre squared.

And I would say that was slightly more efficient.

I would say that was a little quicker and a little easier to do.

So both methods arrived at the correct answer.

But what do you think was more efficient? Which one did you like best? There's no right or wrong answer to that.

But let's do a check.

So this is the flag of Peru.

It consists of three rectangles of equal size.

What is the area of the red part? Think about that and what is the area of the white part.

So you might notice that two rectangles make up the red part.

Okay, well pause the video.

If you've got the opportunity to work with somebody else, please do and I'll see you shortly for some feedback.

Okay, how did you get on? Let's have a look.

You could do this, you could work out the area of the whole flag to start with.

So eight times 12, it's a times table's back so it might have come to you instantly.

That's 96 centimetres squared.

And then if we do 96 divided by three, that gives us 32 centimetres squared.

So therefore each part is worth 32 centimetres squared.

Still not quite answered it yet though.

We've got the white part certainly.

But what about the red part? So the white's 32 centimetres squared.

What do we have to do with those two red parts? Pretty straightforward thing is to combine them together.

32 centimetres square plus 32 centimetres squared equals 64 centimetres squared.

We needed a little bit of thinking, didn't we? It took a few different steps but we got there.

Now you might have also used this strategy which is possibly even more efficient.

And with probably all of the questions you're going to do today, there's at least two different ways to work it out.

So do have a little think.

You might have done this.

12 centimetres divided by three gives us four centimetres.

So we divided that length of 12 centimetres up into three lengths.

So each section's got a length or a width of four centimetres.

So therefore it's going to be four times eight and four times eight and four times eight.

So four times our eight times four is 32 centimetres squared.

That's the white.

And then you could do eight centimetres times eight centimetres is 64 centimetres squared.

Lots of ways to work it out.

Okay, well you may recognise this flag.

This is the England flag.

It's a large one this time.

Notice the dimensions have changed.

The other ones were in centimetres.

They were teeny tiny ones.

These are large ones that you might see maybe outside a big building, something like that.

So it's a five metre by seven metre flag, pretty big.

This time the different coloured sections are not equal to each other.

They were before.

They're not this time.

So the amount of white is different to the amount of red.

We're gonna work out how much white there is, how much red there is using the information that we're given.

Definitely need to do some thinking here.

Still possible to calculate the area taken up by each colour using the dimensions given.

So the area of the whole flag, let's start with that, is five metres by seven metres, which is 35 metres squared.

Now removing the red part, removing the one metre from the length and one metre from the width will give the area of just the white parts.

So we're imagining the flag with the red bits taken away.

It would look a little something like that.

Now we can work it out.

So we've taken one from five and we've taken one from seven and that gives us four by six.

That's the area of the white part.

So four metres by six metres is 24 metres squared.

So that's the white part.

Now I wonder what we can now do to work out the area of the red part.

Now remember we did know the total area of the flag and now we worked out the white part of the flag.

So the red part will be the difference between those two.

The difference between the white section and the whole flag is the area of the red cross.

So therefore, and we can do this by subtracting or counting on 35 metres squared, take away 24 metres squared equals 11 metres squared.

So therefore the red part is 11 metres squared.

And both of those together equaled 35 metres squared, which is the area of the whole flag.

That again, that took a few different steps, didn't it? It took a bit of thinking out of the box as they say.

But we got there in the end.

Let's have a check.

What strategies could you use to work out the red area and the area of the white cross in the Denmark flag? Pause the video, do some thinking, do some talking.

Let's have some teamwork and we'll have some answers shortly.

Pause the video.

How did you get on? A little bit like the England flag, wasn't it, this one? Similar sort of problem I think.

So what could you do? We haven't got any dimensions here or anything like that.

You've just got to think about what strategies you would use.

And you might say something like this, work out the area of the whole flag by multiplying the length and width and then subtract whatever the white width is.

It's not a given and the white length.

So subtract those, that'll give you the area of the red part.

So multiply the new dimensions, the dimensions of just the red part together to give the red area and then subtract the red area from the flag area to give the white cross area.

So very much like the England flag.

You might have said something a bit different to that.

That's one possible strategy.

Time for some independent practise.

So task A, calculate the area of each of the colours in the following world flags.

So you've got Ireland which is split into three equal rectangles.

It's got dimensions of 10 centimetres and 18 centimetres.

What can you do with those numbers and that information I wonder? And for Columbia, again, we've got three different colours, but you might notice they're not equal unlike the Ireland flag.

Both the blue and the red rectangles do have the same dimensions.

Now you might see here we've got three different pieces of information given here.

So three different dimensions, eight centimetres, 18 centimetres and four centimetres.

Have a good think about what you need to do with that information.

And then we've got the flag of Costa Rica.

And here the blue and white rectangles all have the same dimension.

So there's two blue ones, they're the same as each other and two white ones, they're the same as each other and same as the blue.

And then again, you've got lots of information there.

Have a think we need to do with it.

And then finally Finland, which I think looks a bit like the England one and a little bit like the Denmark one.

Reminds me of that.

Can you work out the white area and can you work out the blue area? Have fun with this, enjoy, persevere and I'll see you soon.

Welcome back.

How did you get on with that? Definitely some thinking required there, wasn't there? Let's have a look at some answers then.

So for number one then this is the Ireland flag.

The three rectangles all have the same dimension so we know they're going to give the same answer and I think it wasn't too difficult to work out the area of the flag.

So 10 by 18, that's 180 centimetres squared.

And then if we divide that by three, that will give us the area of each of the colours.

So 180 divided by three is 60.

So the green part, the white part and the orange part each have an area of 60 centimetres squared.

And here is Columbia.

Remember in this one the blue and red rectangles took up the same area as the yellow rectangle.

And we've got information about the yellow one.

We've got enough to work out the area of that already.

So as they're identical to each other, that means you can halve the 72 centimetres squared to calculate their areas or realise that they're each two centimetres wide.

So the area is 18 centimetres by two centimetres, which is 36 centimetres squared.

Bog one if you got that.

Costa Rica.

So the whole flag has an area of 48 metres squared.

You might have noticed that if you swap two of the colours around, there will be three equal rectangles of red, white, and blue.

So divide 48 metres squared by three and you've got 16 metres squared for red, the same for white and the same for blue.

So in my mind I kind of moved the blue up to the top part and the white down at the bottom giving me a flag of three colours, a bit like the Ireland flag.

And Finland, the whole flag area is eight metres by five metres, that's 40 metres squared.

Removing the blue cross would give a white rectangle with dimensions of seven by four, which is 28 metres squared.

And then we look at the difference between 28 metres squared and 40 metres squared, which is 12 metres squared.

So that must be the area of the blue cross.

So white 28 metres squared and blue, 12 metres squared.

Very, very well done if you got those.

They were undoubtedly challenging.

Well done.

Let's move on to cycle B.

That is efficient strategies with compound shapes.

So Jun and Lucas have been measuring parts of the Oak Academy playground, that's the school they go to, with metre sticks to work out their area.

Okay, so you can see that that's like an aerial viewer bird's eye view of what they've been doing.

They're the dimensions.

So the playground's a compound rectilinear shape as you can see.

They start with a play equipment section and Lucas says four metres by 10 metres equals 40 metres squared and seven metres by 10 metres equals 70 metres squared.

Hmm, I think I can see something wrong there.

Can you spot it? Something's gone wrong.

I can see why he's done that there.

And then he says 40 metres squared plus 70 metres squared equals 110 metres squared.

I think the calculations that he did were correct.

I don't think he maybe did the right calculations there or certainly one of them.

So he thinks here is 110 metres squared.

But Jun like me thinks that is wrong.

He does agree with parts of it though, he agrees with four metres by 10 metres equals 40 metres squared.

But it's not seven by 10 metres.

It looks like it, doesn't it, because they're the numbers given.

But aha, can you see? It's not 10 metres.

We don't need the the area of all of that.

It's seven metres by six metres 'cause we have to take away the four metres that we've already calculated from that 10 metres.

So 10, take away four, six, so that's seven times six, which is 42 metres squared.

40 metres squared plus 42 metres squared equals 82 metres squared.

And that is the correct area of that part of the playground.

So well done, Jun.

Jun also knows the shape can be decomposed into rectangles in a different way.

So that's one way to do it.

Can you spot a different way? Well where else could that dotted line go? How about there? Seven metres by 10 metres equals 70 metres squared.

Now the seven and the three together make that 10.

So therefore that part of the rectangle of the shape must be three metres.

So that part must be four by three.

So seven by 10 equals 70, three by four equals 12.

And then add those together.

We've got eight two metres squared and that's the area of the compound shape.

Here's Laura.

She sees it differently once again.

She can see a square with a missing rectangle.

Oh yes, that's clever.

So a bit like this.

So she can see a 10 metre by 10 metre square that's quite easy to work out.

It's a 100 metre squared.

And then she's going to take off the area of this part.

So that must be six metres there and that must be three metres there.

Six times three, it's the times, tables back, is 18 metres squared.

So a 100 metres squared, take away 18 metres squared, gives you 82 metres squared.

Let's do a check for understanding.

This time Jun and Lucas are working out the area of the lower playground.

So it's a different part of the school.

Now you might notice it's a bit different this time.

Again, just like before, it's a compound rectilinear shape.

But this one I could split that into three rectangles and in fact, I could do that two different ways as well.

Discuss with your partner all the different ways that you can think of that would give the area of this part of the school.

So take some real good time to work this one out, have a good discussion and I'll see you soon for some feedback.

Yes, I can definitely see a couple of approaches here.

You might have said this, 11 metres by 11 metres is 121 metres squared and then three metres by two metres is six metres squared.

And then subtract them.

So that'll give you 115 metres squared.

So what you're treating it like a square and then subtracting that little rectangle.

You might have done it differently there.

You might have done 11 by five is 55, 8 by two is 16, 11 by four is 44 and add all those three together and you've got 115 metres squared.

So that's treating it as three rectangles and working out the area of each.

But either way, you get the answer a 115 metres squared.

Very, very well done if you got that.

It was not easy.

Congratulations, you're on track.

Okay, let's have a look at this shape.

It's a rectilinear shape and it's got six squares inside it.

We've only got one piece of information here, which is one of the dimensions, which is 12 centimetres.

So just one of those dimensions is given.

What's the area of the shape? Now it feels like there's not nearly enough information given there to work that one out, but if we do some thinking, we'll find out there actually is.

"I'm going to work out the dimensions of one square starting by dividing by three," so says Lucas.

Okay, I can see where it's going.

So if you take that 12 centimetres and divide it by three, that will give you the dimension of a square, like so.

So that's four centimetres.

Okay, I think that's helpful.

Now remember there's squares, so I think we know another dimension.

Four by four's 16.

So each of those squares has got an area of 16 centimetre squared.

It's starting to come together now, isn't it? What could we do with that do you think with that information? What strategies could we use to work out the whole area now we know that? Well there are six squares.

Yes, I can see that.

And then 16 centimetre squared times six gives us 96 centimetres squared.

But Jun sees it differently.

He says, "I can imagine a bigger square around the shape." Oh yes, okay.

Yes, just like so.

I like that.

And then he is going to work out the area of that what would've been the biggest square.

So 12 centimetres by 12 centimetres is 144 centimetres squared.

So that would be the area of that shape.

Of course it's going to be less than that though.

Now Lucas had worked out that one square had an area of 16 centimetres squared.

So you're going to use that fact and then three of those squares are missing.

And this is why by the way, it's good to work with a partner.

You can use information from each other, you can bounce off each other.

But three of those 16 centimetre squared squares are missing.

So if we do 16 times three, that gives us 48 centimetres squared.

That's the area of the missing part.

So 144 centimetres squared, subtract 48 centimetres squared gives us 96 centimetres squared.

So a different way to work it out, but we've got the same answer.

Did you like that method, I wonder? And then Laura sees a problem differently once again.

So three children have seen it three different ways.

She says, "If we move this square," oh, I see where she's going with this.

She's noticed something.

"Into this gap.

." Yes.

Yes.

What would it give you? A rectangle.

"It would create a 12 centimetres by eight centimetres rectangle." Yeah, I like that.

That's quick, isn't it? An eight by 12.

So times tables back, gives us 96 centimetres squared.

So three very different strategies, all of which reached the final answer of 96 centimetre squared.

So welcome to all three of those children.

Okay, let's do a check.

So all three of the pupils got the right answer, but whose method do you think was the most efficient? So let's remind you again.

So here's we've got Lucas here, he did it in three steps.

He did 12 divided by three is four, four by four 16, 16 times six is 96.

And Jun did it in five steps.

12 by 12 is a 144, 12 divided by three is four, four by four, 16, 16 times three is 48 and 144 take away 48, 96.

And Laura did it in three steps, 12 divided by three is four, four times three is 12 and 12 times eight is 96.

My question to you is, which do you think was the most efficient? Now there is not a right or wrong answer here.

There's certainly one that I found the most efficient, but what do you think? You decide.

Pause the video.

Okay, what do you think? Did you manage to chat about that with the people around you? Did you come up with the same answers or different, I wonder? Well you might have said Lucas and Laura used the same number of steps.

They did.

They both used three steps.

That's correct.

However, all of Laura's calculations were instant recall facts that she knew from her times tables.

Whereas if you look at Lucas's, he had to work out 16 times six.

So I think her three calculations were slightly easier than his three calculations.

So I would say that she just about had the most efficient method.

Let's do some independent practise.

So Jun and Lucas have been measuring the dimensions of the Oak Academy car park.

Find different ways to work out its area, calculate the most efficient.

So you can see we've got a compound rectilinear shape there and we've got some information, but you need to do something with that information.

Number two, calculate the area of this shape in different ways.

Find the most efficient way possible.

This one's in metres.

Pause the video and good luck.

Welcome back.

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

So number one, the area of the car park, the Oak Academy car park, well you could do five times three is 15.

So we are kind of calculating the area of that rectangle on the right, the small one and then 12 by six is 72 metres squared.

And then add them together.

We've got 87 metres squared.

But you might have done it differently and maybe we even did it both ways and compared both for efficiency.

So 11 by 12 is 132.

So we're working out kind of the big rectangle first and then we're subtracting the small one from it.

So five metres by nine metres is 45 metres squared.

And then 132 metres, take away 45 metres squared, gives us 87 metres squared.

Two different ways of getting the same answer.

Personally, I like the first one best.

I think that was the most efficient for me.

You may have used a different strategy altogether.

And then number two, you could do something like this.

10 by 10 is a 100, a 100 metres squared, six by six, you can see a square missing there, is 36 metres squared, a 100 metres squared take away 36 metres squared gives us 64 metres squared.

Again, you may have used a different strategy.

Very well done if you got those.

They were certainly not easy.

We've come to the end of what I think has been a wonderfully challenging lesson.

The theme of the lesson has been calculating the area of shapes made from two or more rectangles.

And we've explored some quite challenging problems, all about that.

With compound shapes made up of more than one rectangle, it is usually possible to calculate the area using more than one strategy.

Sometimes there's two even three ways to do it.

Take the time to see if there's anything about the shape that might lead to a quicker, more efficient way of calculating the area.

You've been really impressive today and I've thoroughly enjoyed spending this time with you.

Hopefully I'll see you again soon for more maths.

But until then, take care.

Enjoy the rest of your day and I'll see you soon.

Goodbye.