Lesson video

Hi there.

My name is Mr. Tilstone.

I'm a teacher, and my favourite subject is definitely maths.

You might already have some familiarity with area and perimeter, and today we're going to be doing an investigation all about those.

So if you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is I know that shapes with the same area can have different perimeters and vice-versa.

So shapes with the same perimeters can have different areas.

Let's investigate that.

Our keywords.

My turn, area, your turn, and my turn, perimeter, your turn.

I think those are words that might already be familiar to you, but it's worth a little recap.

Area is the measurement of a flat surface.

So a 2D surface.

It measures a two-dimensional space.

And the distance around a 2D shape is its perimeter.

Now those terms are easily confused, so make sure you know which is which.

Our lesson today is split into two cycles, two parts.

The first will be same area, different perimeter, and you probably guessed it, the second will be same perimeter, different area.

So if you're ready, let's begin by looking at same area, different perimeter.

In today's lesson, you're going to meet Izzy and Jacob.

Have you met them before? They're here today to give us a helping hand with the maths.

Let's have a look at this tangram.

Have you had some recent experience using tangrams? Hopefully you have.

They're very helpful.

The tangram has been decomposed to form a new shape.

So can you see that? So that tangram that made a square shape, the pieces have been decomposed, and then recomposed as a sort of fish shape.

Some questions.

Do the shapes have the same area as each other? Well, let's have a think about that.

Has anything been added or taken away from the original square tangram shape? No.

Have any of the shapes been duplicated? No.

Have any been enlarged or shrunk or anything like that? No.

They've been rotated, but it's all the same pieces.

So has the area changed? No.

They've got the same area.

So yes, the shapes do have the same area as each other.

And do the shapes have the same perimeter as each other? Hmm, I think I can probably tell just by looking at them whether they have or not.

What about you? But there is a way to prove it, and you might want to do this if you've got some string and you've got a tangram.

So they don't have the same perimeter.

And what you could do is use string and wrap it around the sides of each shape, and that will show the perimeter.

I don't think we have to do that here because I can visualise it.

I can see that the second shape, that fish shape, would take more string.

Here we go.

So that's wrapping it around the square shape and that around the fish shape.

And you can see there there's more string on the second shape.

So the shark shape has got a longer perimeter.

The smaller shapes have been positioned so that more of their sides form the perimeter of the larger shape.

They're not as tucked in as they are on the square shape.

Have you heard of pentominoes? You might have had some experience in the past using pentominoes.

They are shapes made of five squares.

They're polygons with an area of five units squared.

And that's how we read that small two that's hovering above the word units.

That's units squared, or square units.

Have a look at these three examples.

Well, they're non-examples, they're not pentominoes.

Can you see why? Can you explain why? What's the matter with each one? Well the five squares have to connect on a side, and here, look, they're not connecting on a side.

They are connected, are they, but not by a side, by a vertex.

And the same here, they're connected by a vertex.

And then the final one, well they're not connected by vertex or a side.

They're two separate shapes.

So the polygon needs a perimeter.

There cannot be an unconnected square in it.

That's essentially two different shapes, not a pentomino.

Jacob chooses two pentominoes and he puts them together.

What do you know about the new shape? So it could be any of these shapes.

He's chosen two of them and put them together.

What do you already know about that shape? Have a little think about the area of that shape.

The new shape, because it's made of two pentominoes, which have each got an area of five units squared, the new shape will have an area of 10 units squared.

No matter how you arrange it.

Jacob and Izzy's shapes look different and have different dimensions.

So they've done what we said, they've put two pentominoes together.

Can you see they've got different dimensions though.

Jacob's is slightly wider.

These shapes each have an area of 10 units squared, but they have different perimeters.

Now we could use string to prove that if have these physical shapes, but in this case we can also count.

The perimeter of this shape is 18 units, not 18 units squared, 18 units.

It's a measurement of length.

And Izzy's shape has got a perimeter of 14 units.

So its perimeter is slightly shorter.

You can check that if you like.

The pupils in Class 6 have been making shapes from square sticky notes, and you might have some in your class, you might be able to do the same thing.

What do you notice about these shapes? So think about the area, and think about the perimeter of these shapes.

What do you notice? Let's use a stem sentence.

Shape A has an area of hmm units squared and a perimeter of hmm units.

Well shape A has an area of four units squared.

You can count that very easily.

You might not even need to count it, you might be able to subitize it and just see that there's four.

And a perimeter of how many units, you may need to count this, probably do.

It's got a perimeter of eight units.

Shape B has an area of how many units squared? The same, four units squared.

And a perimeter of what? Again, you might need to count this, you probably will, around the outside of the shape, 10 units.

So those shapes have got exactly the same area as each other, but they haven't got the same perimeter, they've got different perimeters.

So we've proved that.

Two shapes can have the same area but different perimeters.

But how is it possible? How is it possible for two shapes to have the same area but different perimeters? What do you think? Could you explain that? Well consider the position of this square.

So if you don't think about that square, if you think about the other squares, they're same shape, it's just that that square is in a different position.

We've changed the colour of that square.

In shape A, two of its sides are on the outside of the shape, and that counts towards its perimeter, because that's a measurement of the outside of a shape.

But in shape B, three of its sides are on the outside of the shape, counting towards its perimeter.

So that's increased its perimeter.

So it's all about how many of the sides are on the outside of the shape.

Decomposing a shape and repositioning its parts can increase or decrease its perimeter.

So we've proved that shapes can have the same areas but different dimensions and different perimeters.

So shape A has an area of hmm unit squared and a perimeter of hmm units.

If you can fill in the gaps on that stem sentence.

And likewise, shape B has an area of hmm units squared and a perimeter of hmm units.

Okay, let's fill those in.

Now the width in shape A is two units, and the width in shape B is three units.

So they do have different dimensions to each other.

Have they got the same area? Well shape A's got an area of eight units squared, you can count that, you might have counted two fours there, and a perimeter of 12 units.

So, if we count around the outside, that's 12 units.

And shape B has an area also of eight units squared and a perimeter of 18 units.

So they do have different perimeters, even though they've got the same areas.

And once again, it's all about how many of the sides are on the outside of the shape.

How many of the sides of those squares that is.

Let's do a check.

True or false? These three shapes have the same areas as each other, but one of them has a different perimeter.

You're going to need to do some counting.

Pause the video and have a go.

What do you think? Was it true or was it false? It's true.

They each have an area of 5 units squared, but A and B each have a perimeter of 12 units, while C has a perimeter of 10 units.

So it's got a different perimeter to the others, even though it's got the same area.

This time Jacob is drawing different rectilinear shapes, each with an area of six units squared, and then checking whether the perimeter is the same or different.

You might have done this in the past, or you might not have done it.

We're going to use dots on the squares, as you'll see.

He starts by plotting those dots in the squares that he wishes to be part of the designs.

So they've gotta be six units squared, they've gotta be connected by the sides.

Let's have a look what he comes up with.

Lots of possibilities here.

He's gone like this.

So that is what his six squares is going to look like.

And on this one he is done the same thing, but he is arranged them differently.

And again they're not connected by vertices, they are connected by side.

So that will count, that will make a polygon.

Then he draws around the outline of his shapes using a ruler, like so.

It's gotta be very accurate as well, stay on the line, and all that kinda stuff.

So he's drawn around the lines that form the perimeter of that polygon, which has got an area of six units squared, or six square units.

And the same here, drawing around the outside of the shape to form a perimeter.

Let's use a little stem sentence.

These shapes each have an area of hmm units squared.

These shapes each have an area of six units squared.

They've got the same area.

Have they got different perimeters? Well we could count.

The first shape has a perimeter of 14 units, and you can count that to check.

And the second shape has a perimeter of 12 units.

And again, you can count that if you like.

But that is the perimeter.

So the area is the same, and the perimeter is different in this case.

Let's do a quick check.

Hopefully you've got some squared paper in front of you.

Using that squared paper, draw a rectilinear shape, just like the ones we've seen, with an area of six units squared.

And your partner's going to do the same, but you're going to do it a little bit secretly, so that you can't see each other's.

So don't look at each other's shapes as you draw.

And then you might want to sit back to back, or something like that, or put a little book in front of your paper, something like that.

Your aim is to try and draw two different shapes with two different perimeters.

If they look the same when you reveal them, have another go until they are different.

Pause the video and have a go.

You might have come up with all sorts of different things.

Here's just one possible example.

These shapes have got an area of six units squared.

That's one, and that's a different one.

Would you agree those are different shapes? Yep.

And would you agree they've both got the same area? Yes.

What about the perimeter? The perimeter of this shape is 10 units, and the perimeter of this shape is 14 units, same area, different perimeter.

I think it is time for us to do some practise.

So number one, using 12 square sticky notes, or square tiles, or anything that you've got that's square-shaped, that you've got quite a few of, create two different rectilinear shapes with the same areas.

So one is going to have the smallest perimeter that you can create, and one's going to have the largest perimeter that you can create.

So see how big you can make that difference.

And then repeat with different even numbers of square sticky notes.

So you could try 14 or 10, or any even number that you like.

And number two, draw as many different shapes as you can with an area of 12 square units, 12 units squared, but different perimeters.

So you might want to use that dots technique where you're plotting your rectilinear shape first, and then draw the outline after.

Remember, they need to be touching on the sides, not the vertices.

Have fun with that exploration.

See if you can prove that shapes can have the same area but different perimeters.

And I'll see you soon for some feedback.

Welcome back.

Did you manage to prove that shapes can have the same area but different perimeters? Well here's the sticky notes one, here's an example of this.

12 sticky notes, so six in each shape, and they've got the same areas, because they've both got six square units, six units squared, and the first shape has got a perimeter of 10 units, and the second has a perimeter of 14 units.

The squares in the second shape are positioned so that the maximum number of sides is on the outside, not the inside of the shape, thus increasing its perimeter.

And number two, draw as many different shapes as you can with an area of 12 units squared, but different perimeters.

So here's some examples.

So they've all got 12 squares inside them, so 12 units squared, but you can see the perimeters are different.

You're doing really well, and I think you are ready for the next cycle, which is same perimeter, different area.

So let's see if we can flip that around and prove that it's still true the other way.

These are pattern blocks.

You might have had some very recent experience using pattern blocks.

Hopefully you have.

Now, what could you say about their areas? So have a look at them, and see what you can notice about the areas.

Now there's all sorts of answers you can give here, but I want you to notice something about the triangle.

There's a relationship between the triangle and the other shapes.

You're going to have to visualise this, but there's a link.

We can use the triangle as a unit to describe the area of the other shapes.

So before we look to square units, this time it's a triangular unit, it's going to be one unit.

So if that's worth one unit, what's the next shape worth? This parallelogram, can you see two triangles within it? So that's got an area of two units.

What about the next shape? Can you visualise it before we put those triangles in, how many triangles would make that shape? Three.

So we can say that's got an area of three units.

And what about this hexagon? How many of those triangles would make the hexagon? Can you draw the lines in your mind? It would look like this.

The hexagon's got an area of six units, so six of those triangles would make that hexagon.

So they are linked, and that can help us in terms of the area.

Now each side of the regular hexagon has a length of one unit.

So its perimeter is six units.

What could you say about the side lengths of the other shapes? And if you've got the pattern blocks in front of you, you might want to put them together.

You might want to see what the link is, what the relationship is between them.

But let's have a look.

The sides of the equilateral triangle are the same length as the sides of the hexagon.

So we can say the sides of the equilateral triangles are one unit in length.

The sides of that rhombus are one unit.

Three of the trapezium sides are one unit, but not all of them.

So can you see which three? And those three are connected at the vertices, but not all of the sides, 'cause if we have a look, that long side is longer than one unit.

How many units can you see that it is? I can visualise it.

The trapezium's longer side is two units.

So remember that, the long side of the trapezium is two units long.

The perimeter of the hexagon is six units and its area is six triangular units.

A triangle is composed of the smaller triangle and the trapezium.

The perimeter is six units.

Its area is four triangular units.

So we can compose different shapes from these pattern blocks.

These polygons have the same perimeter as each other, exactly the same perimeter, but they have a different area.

Izzy composes two different polygons.

What's the same and what's different? Have a look at them.

You might want to do some counting.

You might want to do some visualising.

Think about the areas, think about the perimeters.

Shape B has a greater area, as a rhombus has been exchanged for a hexagon.

Did you notice that? So they're very similar shapes, but the shape on the left has got an extra rhombus, and the shape on the right's got a hexagon instead.

And the hexagon we can see has clearly got a bigger area.

Their perimeters are different.

A is 9 units and B is 11 units.

What if we rearrange the parts of B to make a new shape? What could change? Hmm.

Could we move anything? Could we recompose it? What about if we did that with the hexagon over? What's the difference between the two shapes now? You want to do some counting? Now they've got different areas, just like before, but they've got the same perimeters.

Nothing's changed about the shape in terms of their area, but because we've repositioned that hexagon, the perimeters changed.

So they've now both got perimeters of nine units.

So it is possible.

Let's do a quick check for understanding, shall we? Let's see how we're getting on.

Which of these statements is true? If two shapes have different areas, they must have different perimeters.

Is that true? B, if two shapes have different areas, they must have the same perimeters.

Is that true? And C, it is possible for two shapes to have different areas, but the same perimeters.

So have a good think.

Discuss this with a partner, if you've got one.

Which of these statements is true? Pause the video.

Did you come up with an agreement, a consensus about this? Let's have a look.

It's C.

It is possible for two shapes to have different areas, but the same parameters.

It's time for some final practise.

Just one task here, but it's very open-ended, make as many examples of this as you can.

Make different shapes from pattern blocks that have the same perimeter, but different areas.

Now if you don't have any physical pattern blocks, you can print some out from the Oak website from this lesson.

Pause the video, have fun exploring that.

See if you can find some shapes that have got the same perimeter, but different areas.

Did you manage to prove it? Can two shapes have the same perimeter, but different areas? Well, let's have a look.

For example, the area's different, however, they both have a perimeter of seven units, so the perimeter is the same.

So that's one of many, many examples.

We've come to the end of the lesson.

Today's lesson has been shapes with the same areas can have different perimeters and vice-versa.

So shapes with the same perimeters can have different areas.

It's possible for two, or more, shapes with exactly the same areas, to have different perimeters to each other.

And here's an example using tangrams. Likewise, it's possible for two, or more, shapes with exactly the same perimeters, to have different areas to each other.

I hope you've had fun investigating this topic.

I certainly have had fun teaching it, and I hope to see you again soon for some more maths.

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

Take care, and goodbye.