Lesson video

Hello, I'm Mr. Tilstone, I'm a teacher.

If I've met you before, it's lovely to see you again, and if I haven't met you before, it's lovely to meet you.

Hope you're having a great day and that you've been very successful so far today.

Let's make it an even better day and an even more successful day by tackling this lesson, which is all about area and perimeter and solving problems to do with area and perimeter.

So if you're ready, let's begin.

The outcome of today's lesson is, I can apply understanding of calculation of area and perimeter of polygons to solve problems. And I'm confident that you've got lots of knowledge now about area and perimeter.

Well, let's see if we can apply that knowledge.

Our keywords today, my turn, area, your turn.

My turn, perimeter, your turn.

And I know that you know what they mean, but it's worth a reminder.

It's worth being sure, because those two are very easily confused and very often confused.

So let's have a reminder.

So area is a measurement of a flat surface.

It measures a 2D space, so the space inside something, and the distance around a 2D shape is its perimeter.

So it's a measurement of length.

Our lesson today is split into two cycles.

The first will be area problems, and the second area and perimeter problems. So if you're ready, let's start by really focusing on area problems. And in this lesson you're going to meet Alex, have you met him before? He's here today to give us a helping hand with the maths.

Alex and his class have been learning about calculating area and you might have been doing that too recently.

I suspect you have.

He says, "For rectangles and parallelograms, I can multiply the base by the perpendicular height." Is this ringing bells? Yeah, you knew that.

For rectangles, the base and perpendicular height are actually a pair of sides.

That's true, isn't it, you don't need to look too hard to find them.

For any triangle, the area is base multiplied by perpendicular height divided by 2.

You might have had some recent experience of combining two congruent triangles, so two that are exactly the same and then halving them.

So you find the area of that parallelogram and then halve it to find the area of the triangle.

So it's linked, it's closely linked.

I need to remember to check the perpendicular height measurement is from the base.

Calculating area can be applied to more complex problems. So let's have a look at this example here.

What do you notice about this shape? It's a trapezium.

What's the area of that trapezium? Ooh.

And that's tricky, isn't it? Alex says, "I don't know a formula to calculate the area of the trapezium, I'm going to decompose the shape." Are you familiar with that word, decompose? Is gonna take it apart.

It's gonna separate it into two different parts or more.

Decomposing the shape into smaller shapes where the area can be calculated is a good strategy.

Yes.

So look what happens when he draws this line.

Can you see that it's been decomposed into two shapes that you can calculate the area of.

I can see a rectangle there or a parallelogram if you like, and a triangle.

And we've got strategies for calculating the area of both of those shapes.

Alex now has a rectangle and a triangle and can calculate the area of the parts of the trapezium.

And then of course if he recombines that he's got the area of the trapezium.

He says, "I know some side lengths but I'm going to annotate the diagram with everything I know so far." So have a look yourself.

What side lengths do you know, what's going to be helpful here? Okay, so he's annotated that side length.

It was a known side length.

He already knew that was 8 centimetres and he's annotated this part by labelling it 5 centimetres.

Where's he getting that 5 from? Can you see? Well the 15 centimetre line could be thought of as the 10 centimetre side plus another 5 centimetres.

So 5 centimetres is the length of that triangle side.

Let's have a quick stop and a check.

See if you still know how to calculate the area of a rectangle.

Can you calculate the area of this rectangle? Pause the video.

Did you remember how to do it? Base multiplied by perpendicular height.

We've got both of those.

The side lengths are 10 centimetres and 8 centimetres, 8 times 10 equals 80.

The area is 80 centimetres squared.

So that is the rectangle part.

And now he's going to calculate the area of the triangle.

He is done the rectangle.

Now time for the triangle.

"The base," he says "can be the 8 centimetre side and the perpendicular height is therefore 5 centimetres." It's a right-angle triangle so that side length can count as a perpendicular height.

So 8 multiplied by 5, 8 centimetres multiplied by 5 centimetres equals 40 centimetres squared.

That would be the area of the rectangle.

But 40 centimetres squared divided by 2, or halved, equals 20 centimetres squared, so that's the area of the triangle.

Here we go, now what could we do? We've got the area of the rectangle, we've got the area of the triangle.

What can we do now? He sums the 2 areas to find the area of the trapezium.

He adds them together and that gives us a total of 100 centimetres squared.

So although he didn't know how to calculate the area of a trapezium, he did know how to calculate the area of the shapes that he decomposed a trapezium into.

Let's look at a different problem, have a look before we dive in, see what you notice, what information is given, what information might you need? What might the question be? Let's have a look.

Alex says, "I can use the dimensions of the rectangle to calculate the area of the whole rectangle." Hmm, 14 multiplied by 22 equals 308.

And you might have needed a written method for that.

The whole rectangle has an area of 308 centimetres squared.

So that's a whole rectangle.

Now I wonder what we could be trying to find out here.

He knows the area of the whole rectangle is 308 centimetres squared.

Now he thinks about the isosceles triangle.

So can you see three triangles in there? Two of them are grey and one's are sort of very light purple.

Which one's the isosceles one? Well the two grey ones are both right angle triangles.

But you could also say they're scalene triangles because all three sides are different.

Whereas the very light purple one that's got two sides that are the same length and one that's different.

So that's the properties of an isosceles triangle.

So that's what he's thinking about now, he says, "I can see that the triangle has a base of 14 centimetres and a perpendicular height of 22 centimetres." "It's the same as the length and width of the rectangle." So it is.

So the triangle's area is half that of the rectangle 'cause it shares the same dimensions so it's half of it.

Oh, that means a shaded section must be half the area of the whole rectangle too.

Do you think he's right? I can just halve the area.

Half of 308 is 154.

The shaded area is 154 centimetres squared.

So the isosceles triangle is 154 centimetres squared and the two other triangles combined are also 154 centimetres squared.

Okay, it's time for some practise.

Let's see if you can put those decomposing skills into use.

Number one, each shape can be decomposed into rectangles and triangles.

Calculate the area of each whole shape.

Now if you have a look at those 2 examples, A and B, what do you notice? Well look at A, that's already been decomposed for you, but you'll have to decompose B yourself.

So lots of dimensions are given there.

See if you can work out the area of the rectangle and the area of the triangle and then combine them.

And the same for C and the same for D.

Very different shapes.

Take your time before you dive in.

Have a good think about how they can be decomposed and what the known dimensions are.

And number 2, the perpendicular height of the triangle is 10 centimetres.

Calculate the area of the shaded part of the rectangle.

Do have a good think about that.

Don't dive into it.

If you can work with somebody else, I always recommend that as well so that you can share ideas and bounce ideas off each other and share strategies.

Okay, well good luck with that and I'll give you some feedback very shortly.

Pause the video.

Welcome back, how did you get on decomposing those shapes and working out the area of the triangles and the rectangles? Let's have a look.

So number one, did you annotate the diagram to show what you knew? So you could see all the different dimensions on their looks.

Some were given already, and then some you might have added on.

For example, the base of that triangle and the perpendicular height of that triangle could be worked out from the dimensions that you did know.

So the square part was actually a square number.

So 6 multiplied by a 6 equals 36.

So that's 36 centimetres squared.

And the triangle part, the perpendicular height was 7 and the base was 6.

And multiply those together, it's another times tables, so hopefully that was automatic for you.

That's 42 centimetres squared.

But if you divide 42 centimetres squared by 2, that gives you 21 centimetres squared.

And that's how to calculate the area of a triangle.

We're not quite there yet though, 'cause then we have to add them together.

So 36 plus 21 equals 57.

The area of the whole shape is 57 centimetres squared.

And for B did you decompose a trapezium and annotate the diagram? Here we go.

Here's an example.

So you might have added on that other side length, 19 centimetres, which is the same as the known side length.

And then you might have annotated the base of the triangle as 19 centimetres by using the known side length of the shape, 24 centimetres, and subtracting the 5 centimetres.

All sorts of things you could have done there.

And so the rectangle is 19 multiplied by 5.

It's not the times tables fact, but maybe you had a good mental strategy for that.

I would've done 20 times 5 and then subtracted 5.

So that's 95 centimetres squared.

And the triangle part, that's 19 by 19 to give you the rectangle and then that's 361 centimetres squared.

But then you've gotta have it, you've gotta divide that by 2, and it gives you 180.

5 centimetres squared.

Now we've got some adding to do and you might have needed a column addition to do this, but the area of the trapeze and when you add those two together is 275.

5 centimetres squared.

Well done if you got that, there was some tricky parts in there like the decimal.

And for C, did you decompose the pentagon and annotate the diagram like so.

So here you can see the base of the triangle must be 16 centimetres and the perpendicular height of it must be 8 centimetres because we know that the side length of the shape, one of the side lengths is 10 centimetres.

And another one that would be the side length is 18 centimetres.

So 18 centimetres subtract 10 centimetres equals 8 centimetres.

That's the perpendicular height.

So we've got enough information now to work out the areas of the rectangle and the triangle.

So the rectangle, 16 multiplied by 10.

That's a nice straightforward one isn't it? That's 160 centimetres squared.

And the triangle, 16 multiplied by 8 equals 128 centimetres squared.

That's the area of the parallelogram.

But half that and that gives you the area of the triangle.

So that's 64 centimetres squared.

Not there yet, one last step, add them together, 160 plus 64 equals 224 centimetres squared.

And that is the answer.

So when we break it up into little parts, it's nice and doable.

D, did you annotate the diagram to show what you knew like before? So here are some possible dimensions and side lengths that you could have written on to help you out.

The rectangle was 11 by 2, 11 multiplied by 2 is 22 centimetres squared.

That's a rectangle.

Got two triangles this time.

So 3 multiplied by 6 equals 18 centimetres squared.

But then we have to halve that don't we? To make the triangle area that's 9 centimetres squared.

So the area of the shape, 22 centimetres squared plus 9 centimetres squared plus 9 centimetres squared, and you might have done that by doing 22 plus 18 equals 40.

That's 40 centimetres squared.

Very well done if you've got that.

And number two, the perpendicular height of the triangle is 10 centimetres.

Calculate the area of the shaded part of the rectangle.

So the triangle then is 10 multiplied by 12 'cause we know the base, the base is 12 centimetres, so that's 120 centimetres squared.

But you need to halve that of course to give you the triangle area.

And that's 60 centimetres squared.

The rectangle, we've got the dimensions of that, that's 20 multiplied by 12.

You could do that 10 multiplied by 12 and double it, that's 240 centimetres squared.

Not quite there yet.

What's the last step? You need to do a bit of subtraction, this time it's a little bit different.

So the shaded area, if we take the whole area, that's 240 centimetres squared and we subtract the triangle area, that's 60 centimetres squared, it gives us 180 centimetres squared.

Very well done if you got that, a lot to do there, well done if you got there.

Okay, I think probably ready for cycle B.

And that is area and perimeter problems. The equilateral triangle and the square have different areas but the same perimeter.

Hmm, and you might have investigated before how that's possible.

So each side of the triangle is 8 centimetres.

What's the length of the square? Hmm, we need to break this one down a little bit, don't we? Multiplying the side length by the number of sides will give the perimeter of the triangle.

So let's see if you can do that to start with.

It's a nice straightforward times tables fact.

How many sides does a triangle got? Three.

8 multiplied by 3 equals 24 centimetres.

So that is the perimeter of the triangle.

So therefore the perimeter of the square's also 24 centimetres.

But the question was, what is each side length? So what could we do now, hmm.

What could we do with that number 24, how many side lengths does the square have, 4.

So what could we do with the number 4? Dividing the perimeter by the number of sides will give the side length of the square.

So go on then.

24 divided by 4.

Think about your times tables knowledge, it's 6.

So the side length in the square is 6 centimetres.

So lots of little easy steps I think there.

And what's the area of the square? So we know the side length is 6 centimetres.

So what does that make the area of the square then? Hmm, how can we work out the area of a square? We've only got one of the dimensions, 6 centimetres, or have we? It's a square so therefore another dimension is also 6 centimetres.

So we can multiply 6 by 6 or think of it as 6 squared, 6 times 6 equals 36.

So the area of the square is 36 centimetres squared.

It's got a perimeter of 24 centimetres and an area of 36 centimetres squared.

Alex sketches a rectangle and calculates the perimeter and the area and there's enough information there to calculate the perimeter and the area so why don't you have a go too, so let's think about the perimeter.

How could you calculate the perimeter if you know that one of the side lengths is 10 centimetres and another is 5 centimetres, what could you do? And then what about the area? What could we do with those numbers to get the area? Well for the perimeter we could do a 10 plus 5 is 15, and then double that, that will give us all of the side length so that's 30 centimetres, and then for the area, 10 multiplied by 5 equals 50.

So 50 centimetres squared.

He imagines making a vertical cut 2 centimetres from one side, and he sketches his two new rectangles.

Can you imagine that, a vertical cut 2 centimetres from one side? How would that change the dimensions? What would the dimensions of the new rectangles be? So that's what he's doing, and that's what he ends up with.

He's decomposed that shape into 2 rectangles.

What are the new dimensions? What does he know about the new rectangles? The rectangles have one 5 centimetre side length.

Combined, they have the same area as the original.

Nothing's been taken away, nothing's been added, the area hasn't changed.

They've got the same area as before.

Alex visualises and sketches a rectilinear shape made of the two rectangles.

So he's got to picture it first and then sketch it.

Can you do that? Can you think about a shape that could be composed of those 2 rectangles? Alex starts by annotating his sketch.

So he's added in some of the side length.

So he's got 5 centimetres there.

He knows that.

So therefore another one's 5 centimetres and another one.

So he can see that 5 centimetres side length a few times and he can see that 2 centimetres.

So he decomposes shape and the decomposed rectangle has got a side length of 2 centimetres and he can see that in different places on the new shape.

And he's got that 8 centimetres.

Where's that come from? Where's 8 come from? Well it was 10 centimetres, the side length, but he took 2 centimetres off when decomposing it.

So that's where the 8 centimetres has come from.

The area is still the same as the original rectangle, is still 50 centimetres squared.

Nothing's changed, nothing's been added or taken away.

"These two sides," he says, "sum to 6 centimetres.

And I can use this to find the perimeter." So can you see why they'd sum to 6? So the bottom part of that rectangle as you can see it now is 8 centimetres and then you can see 2 centimetres of that 8 centimetres there on the top part of that rectangle.

And then the rest of that must equal 6 altogether because 2 plus 6 equals 8.

So the perimeter, if you add all of those side lengths up is 36 centimetres now.

So it's longer than before.

And if you like to add that up to check, please feel free to do that, but it is 36 centimetres.

So although the area has remained the same, the perimeter has actually changed because we've repositioned part of the shape.

Alex visualises rotating one of the rectangles to make a new shape and he annotates his sketch.

So he is writing in the side lengths as he can see them.

So it's still the same shape but it's been rearranged.

He says, "The area is still the same as the original rectangle.

I wonder if the perimeter is longer again?" What do you think? See if you can make a prediction and justify it.

Okay, let's do a check.

Can you calculate the perimeter of the new shape? Let's see if he's right or not.

Pause the video.

Let's see, you've got enough information there to calculate the perimeter of the shape.

You might need to annotate a few more of the side lengths in, like that one for instance, that must be 3, because the top side length is 5 and the bottom 8, and then 8 subtract 5 equals 3, and then this one must be 2 'cause we can see it's 2 on the opposite side.

So that's all of the side lengths in and now we can add them up.

And he says, "The perimeter this time is 30 centimetres.

It's the same as the original rectangle, but looks different." Okay, it's time for some practise.

Number 1, in each example, the regular shape on the left has the same perimeter as the square.

Work out the side length and area of each square.

So a bit like we did before.

So start by thinking about what the perimeter of the shape on the left is.

So remember you're working out not just the side length of each square, but the area as well.

See what you notice.

Number 2, sketch a rectangle with a length of 12 centimetres and a height of 6 centimetres.

A, visualise, so picture in your head, decomposing it into 2 rectangles and then recomposing to form different rectilinear shapes.

Sketch 2 different rectilinear shapes using the same rectangles and calculate their areas and perimeters, so just like we did before with Alex.

And then B, imagine cutting the rectangle in half and recomposing to make a rectilinear shape.

What is the longest new perimeter that you can calculate? Have fun exploring that.

If you can work with somebody else, please do, pause the video and I'll see you soon for some feedback.

Good luck.

Welcome back.

How did you get on? Let's find out.

So number 1, in each example, the regular shape on the left has the same perimeter as the shape on the right, work out the area of each square.

So for a, we need to multiply 12 by 3 because that is the number of sides and that's the side length, that's 36 centimetres for the perimeter.

The square's got the same perimeter.

So then we need to divide 36 by 4 in this case to give us one side length, and that's 9 centimetres, that's the side length.

But now for the area, multiply 9 by 9 and that gives us 81.

The area of the square was 81 centimetres squared.

And for b this time we've got a pentagon.

So it's 12 centimetres, but this time multiplied by 5.

So that gives us 60 centimetres.

That's a perimeter of both of those shapes.

So now we need to do 60 divided by 4, and that's 15.

And we can do that by halving and halving again, if you like.

So 15 centimetres is the side length.

Now we need to square it.

15 multiplied by 15.

It's not a times tables fact, but maybe had a good mental strategy for that.

That's 225 centimetres squared.

And then for c, still 12 centimetres for the side length.

But the number of sides has gone up one more time.

And it's a hexagon this time.

So it's 12 multiplied by 6.

It's still a times tables fact, that's 72 centimetres, and that's a perimeter of the square too.

So now we need to do 72 divided by 4 to get a side length.

That's 18 centimetres.

Now we need to square 18 centimetres.

18 multiplied by 18 gives us 324 centimetres square.

A little bit trickier that one maybe used a written strategy for that one.

324 centimetres squared.

And then number 2a, you may have decomposed your rectangle with a vertical cut just like this.

You could make different rectilinear shapes.

And here's one example.

So a 6 by 12 rectangle was decomposed into a 6 by 8 and a 6 by 4 rectangle.

And this rectilinear shape has a same area of the rectangle of 72 centimetres squared.

That's not changed 'cause nothing's been added and nothing's been taken away.

Its perimeter is 40 centimetres, which is 4 centimetres longer than the rectangle.

So the perimeter has changed, even though the area hasn't.

You may have decompose your rectangle with a horizontal cut like this.

So we've cut this one exactly in half horizontally.

The new rectangles are actually congruent.

They're exactly the same and we could recompose them as so.

Did you notice an L shape produced the longest perimeter? So well done if you found that one out, this has the same area as a rectangle, but a perimeter of 54 centimetres, which is 18 centimetres longer than the rectangle.

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

I've really enjoyed today's lesson, it's made me think.

Today we've been solving problems involving area and perimeter.

2D shapes can be decomposed into other polygons, and that's what you've done today.

This knowledge can be used to calculate the area of compound shapes.

Compound rectilinear shapes can have the same area, but a different perimeter.

And that all depends on where you position the parts of the shape.

Well, you have been fantastic today.

So a really big well done and a big congratulations on your amazing achievements in this lesson.

You've done yourself proud and you can breathe a sigh of relief.

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

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