Lesson video

Hi there, I'm Mr. Tilstone and it is a great pleasure to be working with you today on your math lesson, which is going to be all about area.

Now, everywhere we look we can see flat surfaces.

For example, this book has a flat surface, the front cover of it, a football pitch has got a really big flat surface.

So each of those flat surfaces has an area.

And we're going to be exploring that concept in a little bit more detail today.

So if you are ready, let's begin.

The outcome of today's lesson is I can measure area using counting with squares as a strategy.

Our keywords today, which we will explore in a my turn, your turn style, my turn area, your turn.

My turn, square units, your turn, and let's have a little bit of a reminder about those words.

They might be familiar to you, but let's check.

So area is the measurement of a flat surface.

It measures a 2D space.

And square units are a square of a set size used to measure area.

It is a three cycle lesson.

Cycle one will be finding area by counting with squares.

Cycle two will be making square units by combining triangles.

And cycle three will be other ways to make square units.

But if you are ready, let's begin by finding area by counting with squares.

In this lesson we've got three helpers and they are Laura, Aisha, and Andeep, so welcome to those.

Area is the measurement of a flat surface.

It measures a 2D space.

That's a very important generalisation that we're going to revisit.

Let's have a look at an example of area.

So here you can see we've got some money.

We've got some notes and we've got some coins.

Each of them has got a flat surface.

That flat surface is a shape, a 2D shape, for example, the notes have got a rectangular surface.

The coins have got a circular surface.

Each of them has an area.

So the note's rectangular, the area of the rectangle can be measured.

The 2 p coin has a different size and shape to the ten pound note.

It's got a circular face, its area is different.

So the area of that circle is different to the area of the rectangle.

The 10 p coin is circular, just like the 2 p coins, so it's the same shape, but it's a different size.

So its area, the area of that circle is different.

And finally, the one pound coin, it looks a bit like it's circular, but it's not if you look closely, however, it does have a perimeter.

So the face of that coin has got a perimeter and therefore a measurable area.

So some different examples there have different shapes that can be found on real life objects, each of which has an area.

Let's do a quick check, so true or false, the area of a rectangle is found by adding the four lengths.

Is that true or is that false? And I'd like you to justify your answer.

You've got two options for that.

A, area is a distance around a 2D shape or B, area is the size of the space inside a 2D shape.

Pause the video, have a go.

Did you manage to come up with an answer to that? Is it true or false? It is actually false.

If you got true, I think you are thinking about perimeter.

Perimeter, which you might have been learning about quite recently.

But now we're talking about area which is linked but different.

So area is the size of the space inside a 2D shape.

Squares can be used to measure the area of a shape, and in fact they're an incredibly useful shape for measuring area.

So let's have a look at a really quick example.

Got some squares there, six squares.

So the following stem sentence can be used and we're going to revisit this quite a lot.

So this shape has an area of mm square units.

See if you can have a go at that, so what is going to be the missing number there? This shape has an area of six square units, right? I'd like to say that again together please, are you ready? Let's go, this shape has an area of six square units.

Brilliant, and it's really important.

So I'd like you to have a go now on your own after three.

Are you ready? 1, 2, 3, go.

Good stuff.

How did you actually get to that number six though? Hmm, well here's Laura.

Laura says, "I counted the squares individually." Yes, you could do that, couldn't you? 1, 2, 3, 4, 5, 6.

That works.

And here we've got Aisha and Aisha says, "I counted the columns." Oh yes, I see.

So 2, 4, 6.

That's a bit quicker, isn't it, than Laura's way.

I like that.

And here we've got Andeep and Andeep said he counted the rows.

Oh, another good way.

So he went three six.

So three different ways of counting, but all of them arrived at the answer of six square units.

Okay, let's do a quick check.

You're going to pause the video, do some counting, and we showed you three methods there, two of which I think were particularly effective.

This shape has an area of mm square units, pause the video.

Okay, I wonder what counting strategy you used.

Did you do what Laura did? Which was to count the squares individually? It might have taken quite a bit of time.

Did you do what Aisha did, which is to count the columns? Did you do what Andeep did, which is to count the rows? Let's have a look, whatever way you did, you counted, the shape has an area of 30 square units.

So well done, if you got that.

Rectilinear shapes are 2D shapes composed of one or more rectangles.

You're going to be hearing that word a lot.

So rectilinear shapes, 2D shapes with one or more rectangles.

So here's some examples.

All of these could be made by combining rectangles together.

Would you agree? So let's have a look at a rectilinear shape this time.

So not a rectangle, although that is a rectilinear shape, but a different rectilinear shape.

So this shape has an area of mm square units, have a look.

What do you think, can you do some counting? Is there a quicker way to do it? You can still measure the area of this shape using square units.

You can see it's composed of square, so it's got square units inside it.

So this shape has an area of 16 square units.

Laura found an efficient way of counting.

Let's see what Laura did.

She saw two rectangles of five square units.

Can you see those one on the left, one on the right? Yes, I can see those and one rectangle is six square units in between them.

So Laura's done well, hasn't she? She's progressed from when she was counting the individual squares to finding quicker, more efficient ways of counting.

Good stuff, Laura.

Aisha found a different efficient way of counting.

Let's see what she did.

She saw two squares.

Oh yes, a missing square of nine units inside a square of 25 units.

Can you see that? Look at the white part.

That's a square, isn't it? The missing part, that's not part of the shape.

But she subtracted it from what would've been a bigger square of 25 square units.

So she simply did 25 takeaway nine.

That's clever, isn't it? Good noticing.

That's what good mathematicians do, they notice things.

So using hopefully some kind of strategy, but if not, just counter the individual squares would work, determine the area of this shape.

And when you've done it, please use a stem sentence.

This shape has an area of square mm units.

Pause the video and give it a go.

Did you manage to count them all? Did you have a good strategy, a good way of doing it? I could see a rectangle and then three individual squares, I could split it in my mind into two parts and then I counted the rectangle.

I went 3, 6, 9, 12, 15 and then added the three extra bits, 18, I didn't need to count 'em individually.

Maybe you did something completely different, but whatever you did, it's got an area of 18 square units.

So that stem sentence is this shape as an area of 18 square units.

Okay, let's have a look at a different example.

Hmm, I wonder if we can be a bit like Aisha was and notice something about this shape.

Have a good look at it, I've noticed something.

It's got an area of 30 square units.

Now you could get there by counting them individually.

Sure that would work.

But says Laura, "If you move the two squares on the left," so can you see those? So two squares on the left of the shape, sticking out one of the top, one of the bottom, if you move those into the gaps on the right, ah yes, you've got a rectangle.

And then, "You can count efficiently in fives or sixes." So you can count the rows or you can count the columns.

So if we move these two squares into those gaps, this is what happens.

How clever, great noticing Laura, good stuff.

That's given Aisha an idea.

She says, "I can see a rectangle with five squares missing." Ah, "I counted in sevens to get to 35 then subtracted five." I can see that, so imagine instead of being white squares, the highlighted bits were filled in.

That would make it a complete rectangle, wouldn't it? You could count it in sevens to get there.

7, 14, 21, 28, 35, so then she took away the gaps.

So 35 take away five or 35 takeaway three, takeaway two.

Either way works, that's clever.

And that saved her a lot of time, hasn't it? She's not had to count the squares individually.

Okay, let's have a look at this shape then.

And I wonder again if there's something that we can notice about the shape.

That stem sentence again, this shape has an area of square mm square units.

I think Laura's about to notice something.

Do you notice something? Is there something special about that shape? I think there's something about that gap.

Well, it's got an area of 30 square units and as per usual, you could have got to that number 30 by counting the individual squares a bit like Laura was doing at the start.

But you also don't need to do that.

You can be more efficient.

So Laura has noticed something about this Rectilinear shape.

So it's made a rectangle, look.

If you make smaller rectangles, it's easier to count.

Remember by the way, a square is a kind of rectangle.

I didn't even count the squares as Laura.

I used subitizing to just see the amount.

You might remember doing subitizing from when you were younger.

Maybe it's something that you still do, which is where you can see a small number of objects and instead of counting them, you just see the amount there.

So she could see fours, she knew they were going in fours, so she counted the large rectangle and then she counted in fours three times and that got her to 30.

Good stuff, Laura, well done.

Here's Aisha.

Never want to be outdone if you've noticed about Aisha.

She's noticed something as well.

She's noticed and I must admit, this is what I noticed as well, that that square could fit into that gap.

If you move the square up into the space, it becomes a rectangle.

You can then count in fives or sixes.

So yeah, she's right, isn't she? I could count in columns and go 5, 10, 15, 20, 25, 30, and I could count in rows and go 6, 12, 18, 24, 30.

Brilliant and here we go.

That's the rectangle that you would count.

So it's all about visualising and manipulating with your mind.

And it's all about noticing things too and taking that time to notice.

So let's do a check, I think.

So that stem sentence, again, this shape has an area of mm square units.

How might you efficiently count the squares to find the area? 'Cause I don't think you have to go 1, 2, 3, 4, 5, 6, et cetera.

I think there's a better way.

Is this something you can notice? Now I think I can think of at least two different strategies there, but let's see what you come up with.

Pause the video.

Okay, how did you do that? What strategy did you use? I noticed that there was a big square with a little square missing.

So that's what I did, I counted the big square in rows and I counted the little square and then subtracted.

But you might have done something different.

I could also have split it into rectangles.

So yeah, a rectangle on the left, two in the middle, two squares, and then a rectangle on the right.

But either way, either way it's 32 square units.

So that whole stem sentence is this shape has an area of 32 square units.

So this is what Aisha noticed.

It's a six by six square with a two by two hole in the middle.

36 takeaway four equals 32.

Do you think you might be ready for some independent practise? Because I do, so task A one is to complete the stem sentence for each of the shapes.

So we've got that stem sentence there for you to complete.

This shape has an area of mm square units for A, you could count them all individually.

I don't think you are going to need to do that 'cause I think you're gonna notice something or use a strategy.

And same for B.

I can see something straight away that just jumps out at me about B.

It looks unusual shape, but there's something about it.

So this shape has an area of mm square units.

For number two, write and complete the stem sentence for each of the shapes.

So you've gotta write the whole thing this time.

If you can't remember it, just look up and copy the the stem sentence from above and then explain your strategy for efficient counting to find the area.

So very best of luck and I'll see you shortly for some feedback.

How did you get on with that? So task a number one, complete the stem sentence.

This shape has an area of 15 square units.

You might have said something like I skipped counting the columns and counted three groups of five.

And for B, this shape has an area of 15 square units.

You might have said something like, I saw a rectangle of 10 square units and then added five more.

Number two, this shape has an area of 14 square units.

And you might have said something like it's a square with 16 square units, but two of them missing.

And then for B, this shape has an area of 22 square units.

It's like a step shape, isn't it? So therefore it goes seven plus six plus five plus four, and that equals 22.

No need to count the individual squares, but it's got an area of 22 square units.

That's cycle one done, let's move on to cycle two.

Making square units by combining triangles.

Okay, have a look at this shape.

What is difficult about working out the area of this shape by counting squares and is there a solution? Hmm, let's have a look.

Well, I can see squares, but then I can see shapes that are not squares, but I know it's still got an area so it's not quite as straightforward as before.

We can still use the stem sentence.

This shape has an area of mm square units, however, so not all of the shapes are squares.

You can see there, yes, with the triangles.

Two of them are triangles.

The triangles though, there's something special about them.

They're both half squares.

So if we put those two triangles together, it would make a complete square unit.

Let me show you that, here we go.

Take that one away, add it to there.

Now we've got a new shape, but it's got the same area as before.

We've just manipulated it a little bit.

And this shape's got an area of 15 square units.

Now it is all squares.

So we needed to make a small change, but we got there.

Now in that example, we physically moved the triangles, but you don't have to do that.

You can just count them, that would be much quicker.

So let's have a look at this one.

Again, that stem sentence, this shape's got an area of complete mm square units.

Now we're gonna add a little bit to the stem sentence.

Mm extra units are made from combining the partial squares.

So this shape has a total area of mm square units.

So we're gonna count the whole square units first of all.

And then if we put those two triangles together, that's going to make an extra square unit.

So let's have a look at that.

So this shape has an area of 12 complete square units and then one extra unit is made from combining the partial squares.

So this shape has a total area if you add the 12 and the one together of 13 square units.

Okay, so let's think about that strategy then let's combine those triangles.

It's a bit similar to this shape, but it's a bit different as well.

It's got more triangles I can see.

Wonder if you can have a go at completing that stem sentence.

So I'm gonna need a little bit of counting to start with of the square units.

And the square units are in a rectangle shape, so that could be quite simple and straightforward.

And then we're going to count the extra ones.

Okay, how many extra squares will be made by combining those triangles, ready? So this shape's got an area of 10 complete square units and then two extra units are made from combining together those partial squares.

Therefore this shape has a total area of 12 square units.

So even though not all of the units are in a square shape, we can still say how many square units it's got, what the whole area is.

Let's do a check, let's see how you got on with that.

Work out the area of this shape by combining two triangles to make a square.

So can you see it's got lots of individual square units and it's got some triangles that are each half squares.

Use that stem sentence.

Okay, pause the video and give it a go.

Did you get there? Let's have a look, let's have an explore.

So this shape has an area of 14 complete square units.

We could count 14 squares, but we've also got some extra squares made up from combining triangles.

And two extra units are made from combining the partial squares.

Therefore this shape has a total area of 16 square units.

If you got that, brilliant, you are well on track in today's lesson.

And let's put it to the test.

Let's see if you can do some independent practise.

So number one, work out the area of this shape by combining two triangles to make a square, complete the stem sentence.

This shape has an area of mm complete square units.

Mm extra units are made from combining the partial squares.

This shape has a total area of mm square units.

Little tip if you want to put little dots with your pencil inside the ones that you've already counted, you won't count them twice.

And then that's B two, what is the area of this shape? It's a bit of a different kind of shape, isn't it? Big shape, lots of different sides and everything.

And I can see it's made up of squares and triangles.

So what's the area of that shape? And task three, use a blank grid to design your own shape with squares and triangles.

Work out the area in square units.

The final area must be a whole number.

Pause the video, have fun, see you soon.

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

This shape has an area of 15 complete square units.

Three extra units are made from combining the partial squares.

This shape has a total area of 18 square units.

And if you counted up all the whole squares and then combined all the partial squares, that will give you an area of 34 square units.

And then for three, any design really that's got a combination of squares and triangles.

And the final answer, the final area must be a whole number given in square units.

You must use that language, well done if you got that.

And let's move on to the final cycle, which is other ways to make square units.

Let's have a look at this shape.

What's the same and what's different about this shape compared to the others that we've seen so far, particularly the ones in cycle two? Hmm, well it has got some complete squares.

Just like before we could count those complete squares.

It's got triangles, I could see some triangles.

I could see four of them just like before.

But it's something about those triangles that's different.

Did you spot it? They're not half squares this time.

They're different.

So let's have a look at an example there.

This triangle takes up exactly half of two squares.

Just have a look at that, focus on that.

It's in two squares and it takes up half, would you agree? So therefore it's got an area of one square altogether.

If we were to combine those two parts of that triangle together, they would make a square, one square unit.

Let's do a check, so complete the stem sentence.

This shape has an area of mm complete square units.

Mm extra units are made from combining the partial squares.

This shape has a total area of mm square units.

Pause the video and give it a go.

Did you get an answer and did you manage to agree on the answer with the people around you? Let's have a look.

This shape's got an area of eight complete square units.

We can cut eight whole squares and then four extra units are made from combining the partial squares.

So each of those triangles was a one whole square unit.

So this shape has a total area of 12 square units.

Well done if you got that.

Ooh now then, this is a different kind of shape, isn't it? So what's the same about this shape compared to the other ones? And then what's different about it? Hmm, let's have a think.

Just like every shape we've seen so far, it's got complete squares.

It's got lots of complete squares.

However, this time there are no triangles on this shape.

There's something about it isn't there? Did you notice something? Hmm, little clue here.

If you were to manipulate the extra bit on the left and move it into the gap on the right, which you can do with your mind by the way, you don't need to physically do that, it would create a shape of exactly 20 square units.

Hmm, visualise moving part of this shape to create a rectangle.

What is the total area? So have a look at that shape.

It's a bit like the other one 'cause it's got wiggly lines on.

Is there a part of that shape that could be moved into a gap that's the same? And then would that make a rectangle? Pause the video, have a go.

Let's have a look, it's a really unusual shape, isn't it? Well it's kind of an extra bit here that's sort of a wiggly extra bit.

And then there's a gap that's the same, that wiggly extra bit would fit into that gap.

And if you did that, it would make an rectangle with an area of 28 square units.

If you got that, that's very impressive, well done.

Time for the final independent practise tasks.

Number one, complete the stem sentence.

This shape as an area of mm complete square units.

Mm extra units are made from combining the partial squares.

So this shape has a total area of mm square units.

Just have a look what you notice.

I can see triangles, but they're not half squares.

So you're gonna have to do a little bit of thinking about that.

And number two, complete the stem sentence, the same one again, what do you notice this time? Is this something about that shape? Is there a way so that we could look at it in complete square units and give it an exact area in square units? And then number three, write and complete the stem sentence to describe the area of this shape.

So again, if you can't remember it, that's fine, just look at the one above and copy that down.

But again, just notice something about that shape.

There's something about it, it looks very regular, looks very wiggly, but there's something about it.

I think we can turn that into a rectangle.

Okay, pause the video, give it a go, and I'll see you very shortly for some feedback.

Welcome back, how'd you get on with that? Let's have a look.

So number one, this shape's got an area of 10 complete square units and six extra units are made from combining the partial squares.

So therefore this shape has a total area of 16 square units.

So number two, this shape has an area of 30 complete square units.

You could count those, I could count those in five so I can see a rectangle.

So 5, 10, 15, 20, 25, 30, that bit was pretty easy.

But then when we combine the partial squares together, it gives us six extra units.

So two of those triangles combine together to make three square units and that happens twice.

So 36 square units.

And then that stem sentence again, this shape has an area of 24 complete square units.

Six extra units are made from combining the partial squares.

So this shape has a total area of 30 square units.

So we can do that by moving the the wiggly extra bit on the left into the wiggly gap on the right and that would turn it into a rectangle with rows of five.

So 5, 10, 15, 20, 25, 30.

Well done if you got that, that was tricky, that's fantastic.

We've come to the end of our lesson.

Today's lesson was measuring area using counting with squares as a strategy so that all important shape in area is squares and we've been counting with them.

So counting square units is a useful way to determine the area of a 2D shape.

Part squares can sometimes combine to make whole square units.

Counting squares to find area can be made efficient in different ways, including manipulating parts to create a rectilinear shape and then they become easier to count.

I've thoroughly enjoyed exploring using counting with squares as a strategy with you and hopefully I'll see you again soon for another math lesson.

So well done on your efforts and achievements today and take care and goodbye.