Lesson video

Hi there.

My name is Mr. Tilston.

I'm a teacher and if I've met you before, it's lovely to see you again.

And if I haven't met you before, it's very nice to meet you.

Today's maths lesson is my favourite kind of math lesson.

We're going to be doing lots and lots of meaty problem solving challenges.

They're going to take some time, they're going to take some perseverance.

If you can collaborate and work alongside somebody else, I highly recommend that so that you can bounce ideas off each other.

I think if you like a challenge, you're going to really enjoy today's lesson.

So if you are ready, I'm definitely ready, let's begin.

The outcome of today's lesson is I can reason about compound shapes using the relationship between side lengths and area and perimeter.

So let's see if you can recall everything you know about compound shapes.

And our keywords today.

We've got two, my turn, compound, your turn.

And my turn dimension.

Your turn, what did those words mean? Let's have a look.

A compound shape is a shape created using two or more basic shapes.

Is that ringing a bell? And a dimension is a measurement of length in one direction.

Our lesson is split into two cycles.

The first will be calculating missing sides and the second calculating when you know the area.

So if you're ready, let's start by thinking about calculating missing sides.

In this lesson you're going to meet Izzy.

She's here today to give us a helping hand and very good she is too.

This is a compound shape.

Can the area and perimeter be calculated using the information provided? So if you have a look at that compound shape, you might think, well it's got some side lengths missing and it has.

But have we got enough information there? What can be done? Well, it's a compound shape.

So what if it wasn't a compound shape? What could we do? Well, we could decompose it into two rectangles to calculate the area of each.

And you might have had lots of experience recently of decomposing shapes that's going to come in handy now.

So we could decompose it and that's one way that we could do it.

There is another way as well.

We're going to look at that in a minute.

So now we've got two rectangles and you might notice we've got enough information to calculate the area of one of them.

Five times six.

Five centimetres by six centimetres equals 30 centimetres squared.

But what about the other one? What do you notice about that? Can't do 11 centimetres times three.

Because 11 centimetres isn't the side length of that small rectangle.

So we're gonna have to think about what it is.

Is there enough information there to calculate it? There is, you're going to have to do a bit of noticing.

This dimension's unknown but can be calculated using two known side lengths.

Have a look, which ones could be helpful here to work out that missing side length? What about now, is that helpful? It's a difference, isn't it between one of the side lengths that's six centimetres and the other side length, that's 11 centimetres.

We're looking at that difference.

So six centimetres plus something equals 11 centimetres.

Quite straightforward that isn't it? That's five centimetres.

So that was easy when we knew what we were looking for, I think.

So now we've got five centimetres times three centimetres and that gives us an area of 15 centimetres squared.

Not quite got the answer yet though, have we? One last step, what do we need to do? We need to sum together the area of the smaller rectangles.

So 30 centimetres squared plus 15 centimetres squared equals 45 centimetres squared.

Now that same compound shape can be decomposed into two rectangles in a different way.

Have a think about that.

Could we draw the line in a different place to decompose them? What about that? Now what do we know this time, where would just start this time? Because I can see two dimensions that are going to help to work out one of those rectangles.

They're already there.

We don't need to do any working out to find the side lengths.

It's 11 times three, which is 33 centimetres squared.

It's a smaller rectangle that we're going to have to think about a little bit more carefully.

We have got one of the dimensions of the smaller rectangle, but we haven't got the other.

But could we do what we did before? We find a difference between two known side lengths.

What about that? So we've got the difference between three centimetres and five centimetres is two centimetres.

So one of the side lengths of that smaller rectangle is two centimetres.

That gives us a calculation.

Six centimetres times two centimetres equals 12 centimetres squared.

Still not quite there yet are we, we've got one last step.

We've got to combine the areas together.

33 centimetres squared plus 12 centimetres squared equals 45 centimetres squared.

And that is the area of the compound shape.

So a few steps involved there, but if you break them down into a few small steps and tackle each one at a time, you soon get there.

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

What information is needed to find the perimeter this time? So it's the same shape but we are looking for the perimeter.

And you might notice that two of the side lengths are missing at the minute, but can we figure them out? Well this one must be five centimetres because six centimetres plus five centimetres equals 11 centimetres.

We already knew that, that was calculated when we found the area.

There's one missing side length that needs to be determined before the perimeter can be calculated.

Can you see it? It doesn't tell us what the side length is, but we can work it out using two known side lengths.

So it's this side here, what do we know? It's a difference between what and what? The difference between the two known side lengths, that's three centimetres and five centimetres is two centimetres.

So that small side length is two centimetres.

And now we've got enough information to work out the perimeter of the whole shape.

Remember to be efficient.

You can sum them in any order but have a look before you start.

Are there some pairs of numbers that go together really well? For instance that I can see a number bond to 10 straight away that could be a good starting point.

So that's 10, and then I think this strategy of finding pairs of numbers to make one number is helpful.

So what about six and three? That's a nice straightforward one, isn't it? That's nine.

So we can add that on.

And then up to you what you do next, you could add that 11 on, nine and 11 go together quite well, don't they, to make 20.

And then you could add that extra two on, the final two.

Whatever order you do that in, it gives you 32 centimetres.

The area of this compound shape is 45 centimetres squared and its perimeter is 32 centimetres.

We can calculate missing side lengths from known lengths.

We just need to do a little bit of thinking and a little bit of noticing.

Time for a check.

Watch how the area of this compound shape can be calculated.

Watch this.

So we could decompose it into two rectangles and now we've got enough information to work out the area of that rectangle, that's 21 centimetres squared.

And then we've just got to figure out that missing side length.

But we can because it's a difference between seven centimetres and 12 centimetres, which is five centimetres.

And now we've got enough information to work out that area, that's 25 centimetres squared and then we combine them together.

So once again, it is a few small steps.

Okay, can you find another way to calculate the area? Pause the video.

What did you do? How did you approach that? Or you could decompose it this way.

And that gives us one area that's fairly easy to calculate and fact as a times tables fact.

So 12 by three is 36 centimetres squared.

Just gotta figure out the small one now.

Well that's got to be five centimetres hasn't it? Because it's a difference between seven centimetres and 12 centimetres.

So that gives us two centimetres by five centimetres, which is 10 centimetres squared, not quite there yet.

One last step, add them together and that gives us 46 centimetres squared.

So decompose the compound shape into two rectangles.

Work out the one that you've got the dimensions for, figure out any missing dimensions and then work out the area of that one and then add them together.

Let's see if you can put that into practise.

So number one, is Izzy correct? She says I don't need to calculate the missing side.

I can estimate, it's four centimetres.

See if you agree with that.

Number two, calculate the area and perimeter of these compound shapes.

So have a look at A, it's got missing information but you can figure it out.

Same for B, some more missing side lengths but you can figure them out.

Remember we want the area and the perimeter for each one.

And away you go, pause the video.

Welcome back, did you figure those out? Breaking it into small steps definitely works for me.

So number one, Izzy's incorrect.

You can't just estimate.

Often these shapes are not to scale.

And in fact it says this one is not to scale.

She was saying I don't need to calculate the missing side.

I can estimate it's four centimetres.

I know what she means.

It looks a bit like four centimetres but it's not.

It's not to scale, you've got to work it out.

And if you look at these side lengths, the six centimetres to the 11 centimetres, the difference between them is actually five centimetres, not four centimetres.

And two A, this is one possible way of decomposing the shape like so vertically.

And then we've got an area of 55 centimetres squared.

We've gotta figure out a side length here, but we can do that knowing the difference between five and 11 is six.

So that's six by eight, which is 48 centimetres squared.

Add them together and you've got 103 centimetres squared.

You might have done that mentally.

You might have to do a column addition, but either way it's 103 centimetres squared.

And for this one and for the perimeter of that shape, we've got one more side to work out.

We've figured out one of the missing side lengths but we can work it out because it's a difference between eight centimetres and 11 centimetres.

And that's quite straightforward, isn't it? That's three centimetres.

So our missing side length and now we've got enough information to work out the perimeter and whatever way you chose to do that, that will give you a total of 44 centimetres.

And for this one, that's one of two ways to decompose that shape.

You might have done it the other way and that's fine too.

So that gives us one small rectangle with an area of 36 centimetres squared.

That's a times tables effect, three times 12 and then one that we had to figure out.

But we could work out that that's 10 centimetres because it's a difference between three centimetres and 13 centimetres.

We multiply a number by 10, that's 26 times 10, which is 260.

And again, you can hopefully easily add those together mentally, that's 296 centimetres squared.

And in terms of that missing side length, again, we can look at the difference between 12 and 26, that's 14 centimetres.

And now we have got enough information to work out the perimeter by adding them together.

And whatever way you do that, that gives you 78 centimetres, you're doing really, really well.

And I think you are ready for the next cycle, which is calculating with a known area.

The area of this compound shape is 63 centimetres squared.

So just to recap, we know the area this time, what's the missing dimension? So can you see the question mark? One of the dimensions is missing.

Have we got enough information there to work out what it could be, well once again, it's going to take a few little steps.

The missing dimension could be calculated if we knew the area of the rectangle and at the minute we don't.

But we can get there, we can calculate the area of the larger rectangle using the known dimension.

So can you see, we know that's six centimetres by eight centimetres, that's a times tables factor, so hopefully that is automatic for you.

That's 48 centimetres squared.

Now what could we do with that number? 48 centimetres squared, we could subtract it.

We know the whole shape is 63 centimetres squared.

So the smaller rectangle is 63 minus 48, which is 15.

So that means that the smaller rectangle has an area of 15 centimetres squared.

That's still not quite the answer is it? But we've got enough information now to work out the answer.

So what could you do with that number? 15 centimetres squared.

And what could you do with that side length of five centimetres squared? Well we could think of this as multiplication or division.

So in this case something times five equals 15, the missing dimensions there.

Four, three centimetres, three centimetres times five centimetres equals 15 centimetres squared.

So we got there.

Is it now possible to calculate the perimeter? Now that we know that missing side length, is there enough information? What do you think? I think there is.

I think we already know those missing side lengths don't you? Because they're opposite known side lengths.

Opposite sides in a rectangle are equal and that's what these are.

So therefore, there's another six centimetre length and there's another three centimetre length.

Now have we got enough information to work out the perimeter? No, not quite.

We've got one last bit of working out to do.

One last bit of reasoning to do, work out that missing side length.

But we do know two known side lengths and we could find out the difference between them.

The difference between the five centimetres and the eight centimetres.

You could do subtraction or you could add on from five centimetres to eight centimetres.

But either way, it's three centimetres.

Now have we got enough information? Yes, we can add them together.

Remember to look for efficient ways to do it.

They can be summed in any order however you do it.

The perimeter is 34 centimetres.

Let's have a check.

The area of this compound shape is 93 centimetres squared.

What's the missing dimension? Discuss the steps needed with a friend, pause a video, see if you can think of a way to work that one out and I'll see you soon for some feedback.

Did you manage to figure out a way to solve that? Again, lots of little steps are needed for this.

You might have said work out the area of the large rectangle, subtract it from the total area to give the area of the smaller rectangle.

Divide the area of the smaller rectangle by the known dimension of the smaller rectangle.

So that's 63 centimetres squared is the area of the large one.

And the difference between that and 93 centimetres is 30 centimetres squared.

So that must be the area of that smaller rectangle.

And now we've got enough information.

Three times something equals 30, three times 10 equals 30 centimetres squared.

You might have done that using division.

Now there are a lot of steps there.

You don't necessarily need to remember the steps.

What you do need to do is think about the steps, think about what's involved, think about what we know and how we can use that to work out what we don't know.

Time for some final practise.

For each compound shape, use the known area to calculate the missing side lengths and perimeter.

So the first one has got an area of 62 centimetres squared.

What's that missing side length and what's the perimeter? Same for B.

So there's something slightly different about this compound shape.

Look how it's been composed.

Look at the position of the rectangles.

You might need to do a little bit of extra thinking for this one.

And then C, this shape's made from three identical rectangles.

You can hopefully see that, the area of the shape is 105 centimetres squared.

Calculate the missing dimensions and work out the perimeter of the shape.

Looks like there's lots of missing information there and there is, but you can figure it out.

Just take some time to think about it and break into small steps.

Have fun with that and I'll see you soon for some feedback.

Welcome back, how did you get on? So this first shape has an area of 62 centimetres squared altogether.

That's the compound shape's area.

And we can work out the area of this rectangle, that's 27 centimetres squared.

That's a times tables, three times nine.

And we can subtract that from the 62 centimetre squared to give us 35 centimetres squared.

That's the area of the other rectangle.

Now we've got one of the dimensions.

What do we multiply five by to get 35? Other times tables fact, seven.

So that's the missing side length there.

Now can we work out the perimeter? We've still got a little bit more thinking to do, but we are nearly there.

We know those missing side lengths.

They're the same as other known side lengths.

We've just gotta figure out this one.

But we can do that by working out the difference between five and nine.

That's four.

So that missing side length is four centimetres.

And now all we've gotta do is add all of those together in whatever efficient way you chose, the perimeter equals 38 centimetres.

And for B, this compound shape had an area of 88 centimetres squared.

Well, we can do four times 10's, 40, we can do 88 take away 40 is 48.

That gives us the area of the other rectangle.

And then 48 divided by six equals eight, that's eight centimetres.

We're not quite there for the perimeter though, we've got some thinking to do here.

We know those missing side lengths, again, because they're the same as other known side lengths.

Gotta figure out these missing side lengths.

But 10 take away six equals four.

So the two unknown sides total four centimetres.

So it doesn't really matter what exactly each one is.

We know that altogether they're total four centimetres.

So we can use that.

That four centimetres can be added to the known side lengths.

And when we do that, we get a perimeter of 44 centimetres.

So very well done if you got that, that was extra tricky.

And then this shape's made from three identical rectangles.

The area of the shape is 105 centimetres squared.

So we could divide that by three to give us the area of one rectangle.

That's 35 centimetres squared.

We know the length of the two long sides.

So halving that will give us the length of one of them.

It's 14 divided by two equals seven.

So that side length is seven centimetres.

And then seven times something is 35 or 35 divided by seven equals something.

And that is five.

So that's a missing side length, five centimetres.

Now can we work out the perimeter, what can we do? Now we know some of those missing side lengths and we might need to do a little bit of calculation for the other ones.

Okay, well we know those are five or seven centimetres, gotta figure these ones out.

But how about we approach it differently? What must that side length be that's inside the shape? Five centimetres.

So what about if we did 14 centimetres take away that five centimetres to give us nine centimetres, the missing side length, therefore total nine centimetres and we can add that to the known side lengths.

It gives us a perimeter of 52 centimetres.

There was a lot of calculation involved in that and a lot of little steps are well done if you got there to the end, we've come to the end of the lesson and I would say today's lesson has been particularly challenging and I love that.

So well done if you've persevered and well done if you've had some success in today's lesson.

Today, you've been reasoning about compound shapes using the relationship between side lengths and the area and perimeter.

Missing side lengths in compound shapes can be calculated by reasoning, by thinking about what information is already available.

So what can you do with what you know to find out what you don't know? The area and, or the perimeter can then be calculated if they're not already known.

And you've done that lots of times today.

So well done on your amazing accomplishments, achievements and learning successes today, you've been incredible.

I hope I get to spend another math lesson with you in the near future.

It may be about area and perimeter, it may be about something else in entirely.

But until then, enjoy the rest of your day whatever you've got in store, take care and goodbye.