Lesson video

Hi there.

I'm Mr. Tilstone.

It's a great pleasure to be working with you today.

The lesson's going to be all about area, so, if you're ready, let's begin! The outcome of today's lesson, what we'd like you to say by the end of it is, "I can create different shapes with the same area." Our keywords, we've got two of them today, which we'll now explore in a my turn, your turn style.

So, my turn.

Area.

Your turn.

My turn.

Rectilinear.

Your turn.

Now they might not be new words to you.

I'm hoping they're not.

You might have encountered them really recently.

Let's have a look anyway though.

Let's revisit.

So area is the measurement of a flat surface.

It measures a 2D space, and this whole unit and this lesson is about area.

And rectilinear shapes are 2D polygons composed of one or more rectangles.

So a rectangle itself is rectilinear.

And then any shapes that you can think of that could be made of different rectangles are rectilinear.

Our lesson today is split into two cycles.

The first will be rectilinear shapes with the same area, and, the second, non-rectilinear shapes with the same area.

But if you're ready, let's start with rectilinear shapes with the same area.

In today's lesson, you're going to meet these three.

They might not be new faces to you.

You might have seen them before.

Here we've got Jun, Izzy, and Jacob.

So welcome to those three.

They're gonna give us a little helping hand.

Have a look at these different shapes.

Hmm.

What's the same and what's different about those shapes? Have a think.

I can notice a few things that are the same, and I can notice one or two differences too.

Well, first of all, the size of the squares inside the shapes is the same, and that's essential.

That means we can compare them.

They're all rectilinear shapes, so they could all be made from rectangles, including the square which is a kind of rectangle.

But they look different, don't they? Two of them are rectangles.

Some of the dimensions, the lengths and the widths, are not the same.

So they definitely look different to each other.

But, crucially, and this is the theme of today's lesson, they have got the same area, so they're different shapes with the same area.

We're going to use a stem sentence today.

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

So, "These shapes have an area of mmm square units." We're gonna say that together.

Ready? "These shapes have an area of mmm square units." Now just you.

Ready? Go.

These shapes have an area of four square units.

Each of them has an area of four square units, and you might be good at what we call subitizing, which is where you don't actually need to count that.

You can just see that each of them have got four.

But they've got the same area.

So shapes can have some different dimensions but still have the same area.

So that stem sentence again.

"These shapes each have an area of mmm square units." Well, the shape on the left, that's got a width of two units, but, the shape on the right, has got a width of three units.

Despite that, you might notice they've got the same area.

They've both got an area of eight square units even though they've got different dimensions.

So that stem sentence again.

"These shapes each have an area of eight square units." Have a look at these two shapes here.

Look very different, don't they? Izzy says these two shapes can't possibly have the same area.

They look so different.

Well, they do look different, don't they? And Jacob says, "Yes, but count the squares." They both have the same number of purple squares.

I wonder who's right.

Who do you think? Well they're rectilinear shapes.

They're both rectilinear.

They look different, but they have exactly the same area.

They've both got an area of 16, and you can have a little count if you don't believe me.

They've got an area of 16 square units.

So therefore Jacob was right.

So shapes can look really different to each other but have the same area.

Let's use that stem sentence.

"These shapes each have an area of mmm square units." Ready? "These shapes each have an area of 16 square units." Let's do a check.

Let's see how you're getting on.

Which two of these rectilinear shapes have got the same area? So you're going to need to do a little bit of counting.

You might be able to find some efficient ways of doing that rather than counting the individual squares, but, either way, two of those have got the same area.

Can you spot them? And then when you have, I want you to say the stem sentence.

"These shapes each have an area of mmm square units." Pause the video, and give that a go.

How did you get on? Did you manage to agree on an answer with the person next to you, if you've got somebody with you? These shapes each have an area of 15 square units.

So A and C have both got an area, or each got an area, of 15 square units.

So well done if you spotted A and C.

Let's do a quick check for understanding.

I want you to use five square sticky notes.

They've got to be square to create a shape with an area of five square units.

How many different shapes can you create with the same area? Pause the video.

Have fun.

I'll see you soon.

How did you get on with your experiment? How many different ways did you manage to come up with? Did your partner come up with some different ways to you? Well there's lots and lots and lots of possibilities, and here are just three of them.

So those three shapes look completely different to one another, but they've got the same area.

Now these are called pentominoes.

You might have seen pentominoes before, or this might be the first time you've ever seen them, but what do you notice? Something about them.

Pentominoes.

Reminds me of the word pentagon.

Some look like letters or symbols.

Can you see any? I can see like an L shape on one of them.

A U shape on another one.

Some are very thin.

Yes, I can see there's a pink one that's a very thin shape.

Some are very wide.

Yes they are.

There's a sort of yellow one in the bottom middle that's wide.

Some of them are rectangles.

Yes, I can see some rectangles, Again, the pink one.

And some are almost rectangles.

So the one on the far left.

It's nearly a rectangle.

It's just got one square out of place.

Otherwise it would be a rectangle.

But they're all rectilinear.

So whether they're rectangles or not, they're all rectilinear.

They could be made of rectangles.

They all look different.

They do.

Each of these shapes is different to the rest.

But, crucially, they all have exactly the same area.

Did you spot that? And did you spot what the area was? Pentominoes are polygons with an area of five square units.

Now I'm going to show you three examples on the screen that are not pentominoes, and I want you to see if you could spot why.

Have a look at those three shapes.

Why? Why are they not pentominoes? Because they've got five squares.

Well the five squares have to connect on a side, and not all of those do.

Can you see they're connected on a vertex, so that doesn't count.

That's connected on a vertex too.

It doesn't count.

And the polygon needs a perimeter.

There cannot be an unconnected square, but, in the other example, look there is.

So that's actually two different shapes.

It's not a pentomino.

So Jacob chooses two pentominoes and puts them together.

What do you know about the new shape? So it doesn't really matter which of the two he chooses.

He's gotta choose two of them.

What do you know already before he's even chosen them? The new shape will always have an area of 10 square unit.

If he's combining two pentominoes, with an area of five square units, the new shape's got to have an area of 10 square units.

So he's combined them in this way.

What would the stem sentence be? "These shapes each have an area of 10 square units." And he's combined them differently in this case.

Look.

Same two pentominoes.

Completely different shape.

The stem sentence would be, say it with me if you like, "These shapes each have an area of 10 square units," and, another way, a different way of combining those two shapes to make a brand new shape using the same pentominoes, what would the stem sentence be? Ready? "These shapes each have an area of 10 square units." "One of my shapes," says Jacob, "is taller than the others, "but it still has an area of 10 square units." Yes, it does.

The blue part of the shape has been rotated in three different ways.

Yes, it has.

But the total shape still has an area of 10 square units.

Izzy chooses two new pentominoes, and puts them together.

What do you know about the new shape? So she's going to choose some different ones to Jacob.

But what do you know about the new shape regardless of which ones she's chosen? Well, just like Jacob's shape, it will still have an area of 10 square units.

So shall we have a little look? So, as she's chosen these two pentominoes, she's combined them in different ways.

What do you notice? Well, they all look very different to one another, don't they? But what would that stem sentence be? Let's go.

"These shapes each have an area of 10 square units." Well done if you got that.

Jacob and Izzy's shapes look different, and they've got different dimensions.

They're different in lots and lots of possible ways, but the shapes have the same area.

So shapes can look different but have the same area.

So these shapes each have an area of 10 square units.

Let's have a check.

Izzy has made three different shapes.

What can be said about the shapes that Izzy has created.

So, if you've got a partner with you, talk to your partner.

How many different things can you say? Pause the video.

Give it a go.

Okay.

What did you come up with? Did you say something like, "The shapes all look different, "but each one has an area of 10 square units?" If you did, brilliant.

Your response might have varied, but that's the kind of thing that we're looking for.

This time, Jacob and Izzy are drawing different rectilinear shapes each with an area of six square units.

So they're drawing now.

Six square units.

They start by plotting dots in the squares that they wish to be part of their design.

Now remember, just like the pentominoes, they've got to be connected on the side.

So we can't have one kind of on its own, separately, or touching of vertices or anything like that.

But that's one possibility.

So you can see that we've got six different dots, and they're going to be the six square units.

And then Izzy's done a different six dots.

They're going to be her six connected square units.

Then they draw around the outline of their shapes using a ruler, and they're going to be very accurate.

They're going to draw on the lines.

Like so.

Now Jacob's shape's got a perimeter, and therefore it's got an area, and it's got an area specifically of six square units.

And then Izzy's shape.

She's drawing around that too.

She's turning into a shape.

Look how accurate that is by the way.

That stayed on the lines perfectly.

That has got an area of six square units too.

So let's do the stem sentence.

Ready? "These shapes each have an area of six square units." Let's have a check.

So using squared paper, draw a rectilinear shape with an area of six square units.

Use that dot strategy.

I really recommend that.

Your partner will do the same.

Don't look at each other's shapes as you're drawing them.

Your aim is to try and draw two different shapes.

If they look the same, have another go until you've got two different shapes, okay? Pause the video.

Have a go.

Did you manage to do it? Did you come up with two different shapes to each other that each had an area of six units? Well, there are many, many, many possibilities here, and here's just a couple of them.

So that one's got an area of six square units, and that one's also got an area of six square units.

I wonder if you had one of those, or I wonder if yours was different altogether.

I think it's time for some practise.

Task A.

Number one.

Tick all of the shapes which have an area of 10 square units.

So you're going to need to do a little bit of counting.

Hopefully there was some quite efficient counting, if possible, so that you don't need to count all of the individual squares.

We're looking for the ones with 10 square units.

Number two.

Combine the pentominoes to make shapes with an area of 15 square units.

What is the same, and what is different? And if it's possible for you to cut those out and stick those together, all the better.

Have a go at doing that.

And then number three.

Draw as many different rectilinear shapes as you can with an area of 12 square units.

And you might even go onto some new paper after that as well.

You might fill that grid quite quickly and do some new ones.

There are many, many possibilities here, so see if you can get quite creative with it.

But they've got to be rectilinear.

Okay.

Pause the video.

Good luck with that.

Have fun.

See you soon.

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

So the shapes with an area of 10 square units were the ones you can see ticked.

All had 10 square units.

And lots of possibilities for this one.

You might have got something like this.

All three of those shapes had an area of 15 square units.

And here's just a few of the many, many different rectilinear shapes that you could have had with an area of 12 square units.

So as long as when you count there are 12 squares, then you've got it.

And let's move on to cycle two.

That's non-rectilinear shapes with the same area.

So not made of rectangles.

What is the same, and what's different about these two shapes? Let's have a look.

They certainly do look different to each other, don't they? But I think they've got something in common too.

Well they're both non-rectilinear because not all of the angles are right angles as you can see.

We've got some different angles, so it doesn't pass the rectilinear test.

One's a hexagon.

The first one's a hexagon.

The second one's a pentagon.

Both irregular shapes but hexagon and pentagon.

So different shapes.

What they have got in common is the same area.

We're going to investigate that now.

What we're going to do is consider how many complete squares each shape has got and then how many squares can be made by combining the partial squares.

So, in the first example, look, we've got six complete square units, but, if we combine those two triangles together, that makes an extra square unit.

So that would be six plus one making seven square units.

And in the example on the right, we've got four complete square units.

And then if we combine two triangles.

That's five.

Another two triangles.

That's six.

Another two triangles.

That's seven.

So that's also got seven square units.

Let's do the stem sentence.

"These shapes each have an area of seven square units." This time, Jacob and Izzy are drawing different non-rectilinear shapes each with an area of six square units.

Jacob begins by making a table of possibilities.

He knows that two triangles were combined to make a full square unit.

So he knows that if he's got one full square, he'd need 10 triangles.

I don't think it would be possible to draw that.

It'd certainly be very difficult.

If he had two full squares, it would need eight triangles.

Three full squares would need six triangles.

Can you see a pattern forming here? Four full squares would mean four triangles.

Five full squares would mean two triangles, and six full squares would be no triangles.

He's been very systematic about that.

And he's gonna choose a combination, one that looks nice and drawable, nice and doable.

He's gone for this one.

Five full squares and two triangles.

Those two triangles will combine together to make one, giving an area of six square units in total.

So, just like before, he's gonna begin by plotting where the full squares will be, and he wants his full connected squares to be as follows.

So they're connected on the sides.

And then he's going to draw his triangle somewhere around them.

He needs two triangles altogether.

It could be anywhere around those shapes, but he's gonna have one here and one there.

So there's your two triangles making that one square unit.

All he's got to do now is just draw around the outline of the rest of the shape with his ruler, and he's going to be very accurate.

He's going to draw on the lines.

Not in between the lines or anything like that.

He's taking his time.

And there we go.

That's Jacob's shape that's got an area of six square units.

So, Izzy decides to use the same combination as Jacob, five full squares and two triangles but make a different shape.

Let's have a look at that.

She plots her five dots just like before.

There's her five dots.

And she draws the two triangles, but she's going to draw them in different places to before, away from each other this time.

So there's your two triangles.

And then all she's gotta do is draw the rest of the outline with her ruler.

So there we go.

Join them together, and we've got a shape that's got a perimeter and therefore an area, and that area is six square units.

Jun decides to use a different combination to Jacob and Izzy's He's not gonna do five and two.

He's going to do four and four.

So he's going to have four full squares and four triangles.

Let's have a look at that.

Here's his four full squares.

He decided to make a rectangle array of them.

And one triangle's going to go there, and one's going to go there, and one's going to go there and one's going to go there.

It's nice and symmetrical, isn't it, his shape? And all he's gotta do now is join the shape up.

Now he's got a perimeter.

Now he's got an area, an area of six square units.

Different than the other two.

Here they are together.

Look.

All different, but all with the exact same area and all non-rectilinear.

Let's use our stem sentence.

"These shapes each have an area of six square units." Let's have a check.

Which of these combinations will combine to create a non-rectilinear shape with an area of eight square units? Select all that apply.

So your options are six full squares, two triangles.

Will that work? Six full squares, four triangles.

Seven full squares, one triangle, Seven full squares, two triangles.

Have a little think, and select the ones that you think apply.

Pause the video, and I'll see you soon.

What did you come up with? Well, b, six full squares and four triangles.

Those four triangles will combine to make two square units.

That'll be eight.

And also d.

Seven full squares and those two triangles will go together to make one extra square unit making eight square units altogether.

Okay, so tick all of the shapes which have an area of 10 square units.

So look at all of these shapes.

See if you can do some counting, some efficient counting hopefully.

Remember to combine the triangles together.

Which ones have got an area of 10 square units? Number two.

Draw as many different non-rectilinear shapes as you can with an area of 12 square units.

So 12 square units this time.

Got to be non-rectilinear.

So think about using those triangles, okay? Pause the video.

Have fun.

Have an explore.

Be creative.

I'll see you shortly.

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

The shapes with an area of 10 square units were as follows.

And as many different non-rectilinear shapes as you can with an area of 12 square units.

Many examples.

Lots and lots of examples you could possibly come up with.

Here's just two of them.

And you might have done one that didn't have triangles at all.

It might have been something like curved sides.

And that would be acceptable too.

As long as altogether the area is 12 square units.

We've come to the end of our lesson.

This has been all about explaining how to make different shapes with the same area.

Two or more shapes can have the same area even when one or more of them is non rectilinear.

I've really enjoyed today's lesson, and I hope you have to, and I also hope to see you again soon.

But, in the meantime, take care and goodbye.