Lesson video

Hello there, my name is Mr. Tilstone.

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

In today's maths lesson, we're not going to be thinking about numbers at all.

We're going to be thinking about shapes and how they can be decomposed.

If you're ready, I'm ready, let's begin.

The outcome of today's lesson is I know that when a 2D shape is decomposed, so taken apart and the parts rearranged, the areas remain the same.

So we're going to investigate that with two different kinds of resources.

Our key words, well, one's a key phrase really.

So the first one, my turn area, your turn, and the second one, the phrase my turn compound shape, your turn.

Do you know what they mean? Have you encountered those words before? I would think you're probably familiar with area.

Let's have a reminder area is the measurement of a flat surface.

It measures a 2D space.

So you might have had some experience this year or in previous years of finding the area of 2D spaces.

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

So we've got an example here.

Have a look at that.

It's been colour coded just to help you out to see that they're different shapes.

How many shapes that are there? Three shapes.

And they've been put together, composed to make one shape, and that's called a compound shape.

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

The first will be using tangrams to explore area.

Have you heard of tangrams before? And the second will be using pattern blocks to explore area.

Have you heard of pattern blocks? Have you got some in front of you? Don't worry if you don't have those resources.

If you don't, what you need to do is pop to this lesson on the Oak website and print off the additional resources and cut them out and use those instead.

Let's start by looking at tangrams and using those to explore area.

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

Have you met Sophia and Jun before? They're here today to give us a helping hand with our maths.

Sophia has created an interesting shape out of coloured blocks.

Can you see what it is? What do you think? What does it look like? What does it remind you of? It's a chicken.

Now Sophia wants to impress Jun with a fact about her shape.

She says this, I can tell you the exact area of this compound shape.

It's 64 centimetres squared.

Hmm.

So first of all, yes, that is a compound shape.

It's a shape made of two or more different shapes.

It's actually made of quite a few different shapes.

It's quite impressive, don't you think that she knows the exact area? Even though all the shapes are different like that? Jun's impressed too.

He said, how do you know Sophia, hmm? How could she know? Shall we investigate? Shall we find out? The blocks are from an ancient Chinese puzzle called a tangram.

Tangram pieces are special because they can fit together to make a square so they can make all sorts of shapes many, many different shapes when rearranged differently, but they can also make a square.

And when you buy tangrams, they come often in a tray just like that.

What do you notice about Sophia's shape and the tangram? Anything that you notice? Have a look, compare the two.

There's tangram, that's square.

Have a look at Sophia's shape.

What do you notice? Oh, they are composed, made up of the exact same shapes.

So if you have a look, every shape that's in the tangram also features in Sophia's chicken.

There are no shapes missing.

There are no shapes duplicated.

The exact number and the exact types of shapes are present in both.

They've just been rearranged, rotated, recomposed, et cetera.

What do you notice about the shape of the tangram when all of the pieces are arranged like this? What shape is that? That's a nice easy question, isn't it? It's a square.

They form a square.

And then Sophia says, I just worked out the area of the square.

That's how she knew that her shape, her chicken had the same area.

So if this is 8 centimetres in length and it's a square, 8 centimetres times 8 centimetres equals 64 centimetres squared.

So the chicken must have the same area, 64 centimetres squared.

No pieces were taken away.

The size of the pieces did not change.

The area of the shape crucially did not change.

Now Sophia says this, I've made a cat that has the same area as the square.

So she's using those tangram pieces again and she's saying they've got the same area as each other.

Have a look.

Take your time, do you agree? What can you notice? She wasn't correct, but she is now, it was missing a piece.

It was missing the orange piece, the parallelogram, which she's used now as a tail for the cat.

So now she's correct.

So well done Sophia.

Let us have a check.

Jun says, I've made a compound shape that has the same area as the square, it's a shark.

Is he correct? Pause the video and have a look and see what you can notice.

What do you think? Has he made a compound shape? No it hasn't.

It's got the same number of pieces as the tangram, but it's not been arranged to make a compound shape.

Do you know why? The triangle is touching the rest of the shaper to vertex rather than at a side so it doesn't make a compound shape.

In fact, what that's made is two shapes.

Now we've sorted it, now we've fixed it, now it's touching a to side rather than the vertex.

And now it's a compound shape.

And now it's got the same area as the tangram.

It's time for some practise time for you to have a little play with these tangrams. So you are going to get your tangram, either a physical version or one that you've cut up and you're going to decompose that square.

So that starting position for the tangram to make as many different compound shapes as you can with the same area.

So so far we've seen examples like the cat.

It doesn't have to be an exact object or animal or anything like that.

It can just be a random shape, but it's got to have the same area.

So you need to use the same pieces.

You can rotate them, you can reposition them, all sorts of things that you can do.

But do make sure that they're touching at the sides, not at a vertex.

You may wish to draw around some of your new shapes in your maths books.

That's up to you.

That's up to your teacher.

But have fun and I'll see you soon for some feedback.

Welcome back.

Did you have fun experimenting with those tangrams and decomposing them? So what we were looking at there is that you can make lots and lots of different shapes that have got exactly the same area by positioning the shapes differently, by rotating and so on.

Now there are many compound shapes that you could have made with the same area as the original tangram, and some of them may even resemble familiar images if you are feeling very, very creative.

So we've got a shark shape there and a swan shape there.

Are you ready to move on to the next part of the lesson? So we're done with the tangrams. Now we're going to start thinking about pattern blocks.

And just like before, if you don't have physical pattern blocks, you need to cut some out, print some and cut some.

They are on the website.

Pattern blocks are a set of shapes with a set area.

So here's an example of pattern blocks being used.

The same pattern blocks can be used to create different compound shapes with the same area as each other.

So here's an example of a compound shape that can be made with a certain set number of pieces and type of pieces.

Those aren't all the pattern blocks.

There are different ones too.

That's just some.

So in this case, look, it's used one hexagon, four equilateral triangles and one trapezium.

And that's the shape that we've made.

Jun says he has decomposed the compound shape on the left to create the compound shape on the right.

And he says the shapes have the same area.

Is he correct, hmm? Pause the video.

Have a good, think about that and see what you can notice.

You might even want to recreate these shapes if you like with your pattern blocks.

Pause the video.

What did you notice? Anything? Well, the shape on the left had an extra trapezium that the one on the right hasn't got.

So you can't say they've got the same area.

Everything else about them is exactly the same apart from that.

So they haven't got the same area.

We could add one in and we could make it have the same area, but at the minute it doesn't.

The pattern blocks have a proportional relationship.

So if you can see here, three of the triangles fit into one of these trapeziums to make the same shape.

And that's what Sophia's noticed.

They've got the same area as each other.

You could say the triangle is one third the area of the trapezium.

And again, another example look at the two triangles and the parallelogram, two triangles, Sophia notices have an equal area to the parallelogram.

Yes, they do.

So a triangle and a parallelogram have the same area as a trapezium.

So the shapes are linked in terms of their sizes.

The two compound shapes have the same area.

So let's have a look.

Look at those two compound shapes.

They're made up of different shapes, but they've got the same area.

They've both got one of those parallelograms, which we can see here in blue.

And the one on the right's got one of the equilateral triangles.

The shape on the left has got three more equilateral triangles, but they've made the exact same shape as the trapezium.

So although they look very different to each other, they've got the same area.

And that's what Sophia's noticed.

I could replace three triangles with a trapezium.

My shapes look different but have the same area.

They take up the same amount of space.

Yes they do.

Let's have a check, explain why these compound shapes have the same area.

Pause the video and have a good think.

Well, they certainly look very different to each other, don't they? But they do actually have the same area.

A hexagon has an equal area to two trapeziums and a parallelogram has the same area as two triangles.

The area is equal.

You could rearrange shape B onto A.

So they do in fact have the same area.

It's time for some more practise.

Use pattern blocks to make as many different compound shapes as you can with the same area.

Remember to cut your own out if you don't have any, you may wish to draw around some of your shapes in your books.

And number two, use these four pattern blocks to make the compound shape shown below, which is composed of two hexagons.

So can you see that the outline of it's been drawn, it's made of two hexagons.

How many other compound shapes can you create with the same area? And can you draw them on the grid on this special triangular paper? Good luck with that, enjoy that, enjoy exploring, and I'll see you soon for some feedback.

Welcome back.

How did you get on? So number one, did you make different compound shapes and notice that they had the same area? So in this case, we've actually used the same shapes but rearrange them so they've got a different area.

Did you use the pattern blocks to decompose larger ones? In this example, we replaced a trapezium and the parallelogram with five triangles.

The area of the compound shapes is the same.

So different shapes have been used, but the area is the same.

And you could check that by replacing the shapes.

So well done if you've got some examples like that.

And number two is one possible way to compose a double hexagon shape using the pattern blocks and some other compound shapes with the same area that you might have drawn.

So the shape on the left, that's a double hexagon that was already drawn in there.

That's one way to compose that.

But the shape in the middle, that's a different shape, but it's got the exact same area, uses the same shapes.

And likewise, the one on the far right, it's a different shape, you can clearly see that but it's got the same shapes that make it up.

So it's got exactly the same area.

You might have even worked out what that area was in terms of the number of triangles as a little bonus.

Here's the outline of that shape and here's the outline of that shape.

And now you can count those triangles.

We've come to the end of the lesson, today's lesson has been learning that when a 2D shape is decomposed and taken apart and the parts rearranged, the areas remain the same.

Tangrams and pattern blocks can be used to prove that a shape can be decomposed and then recomposed and still have the exact same area.

So we've got a nice example here.

This is back to the tangrams. You've got that square shaped tangram and it's got a certain number of blocks and a certain size of each block.

And then those exact same blocks have just been rearranged to form two new shapes.

Has the area changed? No, it hasn't.

All three shapes have got exactly the same area, even though they look really different.

I hope you've had fun in today's lesson, experimenting with the tangrams and the pattern blocks and seeing how shapes can be decomposed and recomposed without changing the area.

Well done on your achievements today.

Enjoy the rest of your day, whatever it is you've got in store, take care and goodbye.