Lesson video

Hello there.

My name is Mr. Tilstone and it is my great pleasure and delight to be with you today teaching you a lesson all about area.

So if you are ready to begin, let's go.

The outcome of today's lesson is "I can choose the most efficient way to decompose a shape into rectangles in order to calculate the area." Our keywords today or keyword, there's only one, my turn compound, your turn compound.

Have you heard that word before? Let's explore the meaning.

A compound shape is made up of two or more shapes and they can be any shapes but today we're going to be focusing on rectangles.

Our lesson is split into two cycles.

The first will be decomposing shapes in different ways to calculate the area and the second choosing the most efficient way to calculate area.

If you are ready, we'll start with decomposing shapes in different ways to calculate the area.

Let's go.

In this lesson, you're going to meet Lucas, Sam, and Alex, you may have met them already.

They're going to be there to give us a helping hand.

So Lucas, Sam, and Alex are looking at this compound shape.

You can see it's made up of two rectangles.

It does not have a line splitting it into rectangles however.

So "you could split it into two rectangles using a horizontal line, like so" says Lucas.

Let's have a look at his line that's horizontal.

There we go, yeah.

Now you can see a bit more clearly that it's a compound shape, that it's got two rectangles.

You could split it into two rectangles using a vertical line, like so.

So let's have a look at a vertical line.

Here we go.

Can you see two different rectangles this time? It's the same shape but split into two different rectangles.

So sounds right.

Let's do a quick check for understanding.

Sketch this shape.

So just quickly, not too accurately sketch that shape, split it into two rectangles using a horizontal line, a horizontal line.

And then do the same thing again, but this time using a vertical line, a vertical line.

Pause the video and give that a go.

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

Well, we could split it like so using a horizontal line.

Now you can see two rectangles within that compound shape and the vertical line would go just here.

And you can see again it's split into two rectangles, two different ones.

Now Alex notices something else altogether.

Can you see what he's noticed? How about now? "I see it" he says "as a sort of large rectangle with a small rectangle missing." Can you see that too? So we've got a generalisation here.

I'll say it, we'll say it together and then you say it.

The area of a compound shape made of two rectangles can be calculated by adding the area of each rectangle.

Let's say that together, "the area of a compound shape made of two rectangles can be calculated by adding the area of each rectangle." Now just you.

That may not be the first time you've heard that generalisation.

So they measure the dimensions of the rectangles and they work out the total area.

Let's have a look at that then.

So this rectangle measures seven centimetres by two centimetres.

Can you remember or can you think what you need to do with those two numbers? We multiply them together.

So that gives us an area for that rectangle of 14 centimetres squared.

Now we're going to look at the other rectangle and just like the small rectangle, it's in our times tables.

So it should be an automatic fact for us.

We've got three centimetres by 11 centimetres.

Multiply those together and we've got 33 centimetres squared that still hasn't told us the total area of the compound shape though, has it? What do we need to do with those two numbers? We need to add them together.

33 centimetres squared plus 14 centimetres squared equals 47 centimetres squared and that is the area of the compound shape.

Now we can split it a different way and that's what we're doing now.

So it's still the same shape.

Now this time our dimensions are five centimetres and seven centimetres.

Multiply them together and we've got 35 centimetres squared.

And the small rectangle here on the right, that's four centimetres by three centimetres.

Multiply those together and again, it's the times tables fact so it should be automatic is 12 centimetres squared.

What do we need to do with those two areas? We need to add them up.

When we do, we get the area of the compound shape, which is 47 centimetres squared.

Okay, what's going on here? Can you see this? This is what Alex has come up with.

So remember he saw a rectangle with a smaller rectangle missing.

Now the large rectangle's got an area of, well it's got dimensions, I beg your pardon, of 11 centimetres by five centimetres.

When we multiply those together, we get 55 centimetres squared.

So that's the area of what would be the rectangle, but we know it's not a rectangle and we can take off the little rectangle in the corner to make the compound shape that's got dimensions of two centimetres by four centimetres.

So pretty straightforward, hopefully.

Two times four is eight.

So that's got an area, that gap there of eight centimetres squared.

Now what do we do this time? Are we going to add them again? No, because the gap's missing, isn't it? So what shall we do? Subtract.

55 centimetres squared, subtract eight centimetres squared equals 47 centimetres squared.

So you might notice again it's given us the same area but we've worked it out differently so we have some different strategies there.

So the area of this compound shape can be worked out in different ways, three different ways in this case, giving the same area each time.

So let's have a look at that again.

And there we go.

So each time we've got an area of 47 centimetres squared.

33 plus 14 equals 47, 35 plus 12 equals 47 and 55 take away eight equals 47.

Let's do a check.

Again, sketch, or if you wish, describe three methods that could be used to calculate the area of this compound shape.

Okay, pause the video.

How did you get on? Did you find all three? If you found one, brilliant.

If you found two brilliant.

But if you found all three, amazing, you're right on track.

Let's have a look.

You could split it into two rectangles using a horizontal line and then calculate the two areas and combine.

We don't know the dimensions so we can't say what the exact area is, but that would be our strategy.

So that's a horizontal line.

What about a vertical line? Where would that go? Just here.

So you could split it into two rectangles using a vertical line and then again calculate the two areas and combine.

But there's one more method and it's to draw a rectangle around the shape.

Subtract the area of the small gap rectangle from the large rectangle so that will be three ways of working out that shape.

Time for some independent practise I think.

Okay, so for task one, calculate the area of this compound shape in three different ways.

So you might notice that is the same shape each time.

Dimensions have been given.

You're going to need to draw lines on A and B to split them into rectangles and then add them together, add the areas together and that will give you the area of the compound shape.

And then number two, a square has been cut out of a larger square.

The square that was cut out was half the length of the largest square.

What is the area of the remaining compound rectilinear shape? Pause the video.

Good luck.

If you've got somebody to work with, I would recommend doing that and I'll see you shortly with some feedback.

Welcome back.

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

So for number one then we could split A like so using a horizontal line and that gives us five times three is 15, four times two is eight, 15 plus eight equals 23 centimetres squared.

So that's going to be the area each time.

But let's see some different ways of getting there.

We could use a vertical line and that's what we've done here.

So seven times two is 14 centimetres squared, three times three is nine centimetres squared.

And add those together and we've got 23 centimetres squared.

And for C we drew a rectangle around the compound shape.

The rectangle, the large rectangle had an area of seven by five, which is 35 centimetres squared.

And the small gap that would be a rectangle has got dimensions of three centimetres and four centimetres.

Multiply those together and we've got 12 centimetres squared.

So 35 centimetre squared take away that 12 centimetre squared gives us 23 centimetres squared.

And number two, the area of the remaining compound rectilinear shape, 100 centimetres squared, take away 25 centimetres squared gives us 75 centimetre squared.

So just to go over that, so we've got a square, so that means the dimensions are the same, the length and width will be the same.

And we know that one of the dimensions is 10, so it's going to be 10 by 10.

That gives us the 100, that's what the largest square would be.

And then it's got a square inside it that's been taken out of it.

And that was half of it.

So half of ten's five.

So both dimensions are five.

Five times five is 25 and that's how we get that.

So 100 take away 25 equals 75, 75 centimetres squared.

Very well done if you got that, you had to use your brain a little bit for that one, didn't you? You needed to do a little bit of thinking for that one.

You might have done it a different way, however you might have thought of it as three squares.

Now if I drew a horizontal line across the middle of that shape, that would give me three squares and five times five would give us the area of one square.

So that's 25 centimetres squared and we've got three of them so 25, 50, 75.

That's a different way of doing it.

Very often in fact, almost always with area problems, there's more than one way to work out the area of a compound shape.

Well done so far, you've been amazing.

Let's move on to cycle two, which is choosing the most efficient way to calculate area.

Here's Lucas.

Lucas decides to work out the area of this rectilinear shape by splitting it into rectangles so let's have a look at what we notice about that.

First of all, well it's a rectangle.

It would've been a rectangle if not for the rectangle that's been taken out of it.

So it's like a rectangle with a rectangular hole and he's decided to split it like this.

So now you can see it's got more than one rectangle.

How many has it got? Let's see.

One, two, three, four.

So there's four rectangles in it and then there's the one gap rectangle.

Two if you want to include that.

So Lucas would have to work out the area of those four rectangles.

He says "I can think of a different way to do it as well." Can you? He's used vertical lines there.

What if he used horizontal lines? What would that look like? It would look like this.

How many rectangles now? Let's see.

One, two, three, four.

So four green rectangles this time just like before.

So he would still have to work out the area of four rectangles, however he splits it.

Alex thinks he has a more efficient way to work out the area.

Well let's have a look at that, shall we? "You did it in four steps, Lucas" he says "I can do it in three." Ooh, confident, let's see.

He says "work out the area of the white gap rectangle." Mh-hm.

Work out the area of that, okay.

Yep, you need the dimensions of course, but yeah, okay with you so far.

And then work out the area of what would've been the green rectangle.

Yeah, again, it needs dimensions, but I see that.

And then what do you think he's going to do? Subtract, of course.

He's going to take away, subtract the white rectangle from the green rectangle and that would give the area.

That seemed quicker, didn't it? That involved fewer steps, that involved finding the area of fewer rectangles.

So Alex has used his ruler to measure the dimensions he needs to calculate the area of the shape.

So we've got here one of the dimensions is eight centimetres and the other is 10 centimetres.

So using Alex's method we'd need to work out the larger rectangle and that would give us eight times 10 is 80, so that would've been a rectangle of 80 centimetres squared.

And then the smaller rectangle that we're taking away, it's actually a square, which is a kind of rectangle.

So three centimetres by three centimetres is nine centimetres.

There we go.

So the 80 centimetre squared is a large rectangle.

Nine centimetres squared is the gap rectangle.

What's he going to do now? It's his final step.

Can you remember? Subtract.

So 80 take away nine equals 71.

So the area of that whole shape is 71 centimetres squared.

And I would say wouldn't you that that is more efficient? That seemed quicker, that seemed to have fewer steps.

Alex knows that he could calculate the area of the shape by splitting it into rectangles.

If you can see rectilinear shape, a rectilinear compound shape, you can split it into rectangles and work out the area of each and you could do that with this one.

Sure.

However, he noticed there's something about the shape.

Have you noticed it yet? Because I can see that you could split it into rectangles using horizontal lines.

And I can see that you could split it into rectangles using vertical lines, but I've noticed something else.

There's something special about how this shape looks.

Have you spotted it? Hmm.

"The top rectangle" he says "has exactly the same dimensions as the gap in the big rectangle." Ah yes.

Can you see that? So here that rectangle would fit exactly into that gap.

And what would that turn it into? That would turn the shape into a rectangle.

So a rectilinear shape still, but just a rectangle.

And that would mean we'd only need to multiply the length and the width.

"All I need to do is work out the area" he says "of the large rectangle." Well done Alex, you're quite right.

So he measures the dimensions of this shape that will be needed to calculate the area.

So he's not going to measure all of them because he doesn't need to do that.

He knows he needs to measure the length and width of what will be the new rectangle.

So one of the dimensions is four centimetres, that's four centimetres, that's three centimetres and that's four centimetres.

What do we need to do, do you think with those three numbers? Hmm, well if we add them together, that will give us the length of the new shape.

So four plus three plus four equals 11.

So just picture that three centimetres line just coming down and being in line with the four centimetre lines to make one big line.

So 11 centimetres.

So we've got four centimetres for one dimension, 11 centimetres for the other dimension.

What do we do with those numbers? Four and 11? We multiply them, of course, that's how we work out the area of a rectangle and that's what that is.

So 11 centimetres by four centimetres gives us 44 centimetres squared.

Now that was efficient, wasn't it? That saved a lot of time because otherwise we'd have had to work out the area of a few different rectangles so I like that a lot.

So take the time to notice if there's something special about a shape, take the time to think "Can I save myself time here?" That's what good mathematicians do.

They do the least amount of work that they need to do.

Okay, it's time for a check.

You're going to calculate the area of this shape.

Now I've given you a little hint here.

Look at the dimensions of the extra rectangle on the right and the gap inside the shape because at the minute there's a lot of information on that shape, you may not need all of it.

So pause the video.

If you've got partner to work with, I would recommend that.

and I'll see you shortly for some feedback.

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

Well, two centimetres plus two centimetres plus two centimetres is the length of the new shape.

So six centimetres is the length of what the new shape will be when we move that extra square or rectangle we should say, into the gap.

We don't know if it's a square yet for sure.

So when we do that, we already know the other dimension is five centimetres so therefore six centimetres times five centimetres equals 30 centimetres squared.

Simple as that.

Task B, number one, calculate the area of these compound rectilinear shapes.

So let's have a look.

Let's see some of the things we can notice.

So with A, I can see that you could split that into three rectangles.

Maybe you want to do it a different way there.

Maybe you'd like to draw a rectangle around it and then subtract the gap.

It's entirely up to you.

B's a bit unusual looking, isn't it? It is tilted, but it's still a compound rectIlinear shape.

Have a look.

Is there anything that would make that one easier to work out? How could you do it? How could you split it? This, I can think of a few different ways for that.

Number two, calculate the area of these compound shapes in the most efficient way possible.

So let's see what we can notice about that.

So for A, it looks like a square, but it's not quite, it's a rectangle.

It's an eight by seven rectangle, but it's got a square missing from inside it, a four by four square.

So how could you do that in the quickest, most efficient way possible? And let's have a look at B.

Notice anything about that? I think I do.

Don't want to give it away, but have a look.

Take a few seconds, you'll see something about that.

Something will jump out at you that will make that easier to work out.

All the information that you need is already there.

Pause the video and best of luck.

Welcome back.

That was your final task of the day.

How did you get on with it? So one, A, lots of ways you could do this, but you could do seven centimetres times nine centimetres equals 63 centimetres.

That's what the big rectangle would be.

And then three by three is nine.

That's what the small square is that's missing.

And then 63 take away nine equals 54.

So 54 centimetres squared.

And for B, that tilted shape, the bit sticking out has the same dimensions as the bit missing.

It's just in a funny place.

So we could kind of move it over to make a rectangle.

And then the rectangle would have dimensions of eight centimetres and 12 centimetres.

And that's one of our times tables fact.

So hopefully that was an instant recall for you.

Eight centimetres times 12 centimetres equals 96 centimetres squared.

Well done if you got that.

Here we go.

And number two, calculate the area of these compound shapes in the most efficient way possible.

So we could do seven times eight.

That would give us the area of the large rectangle.

It's the time tables fact.

It's 56 centimetres squared.

Four by four is the square that's missing.

So that's 16 centimetres squared.

And then we need to subtract that small square from the rectangle.

So 56 centimetres squared, take away 16 centimetres squared equals 40 centimetres squared.

And then for B, what we could do is move that rectangle at the bottom into the gap and that would give us a rectangle with an area of 12 centimetres by five centimetres, it's a times tables fact once again.

So important to know those times tables, isn't it? And that is 60 centimetres squared.

If you got that, brilliant.

And we've come to the end of the lesson.

Today's lesson has been choosing an efficient way to decompose a compound shape to calculate the area.

Rectilinear compound shapes can often be decomposed in more than one way.

Sometimes this means splitting them into different rectangles, but sometimes other skills can be used such as subtraction.

So in quite a few of those examples, drawing a rectangle around it and then subtracting the gap was actually more efficient.

So take time to notice if there's anything special about a shape that might mean its area can be calculated more efficiently because that could save you time in the long run.

It's been a real pleasure working with you today on this math lesson.

You've been amazing and I hope to see you again soon for more maths.

But until then, take care.

Have a great day and goodbye.