Lesson video

Hello there, my name is Mr. Tilstone, and I'm really delighted to be working with you today on your maths lesson.

Today's lesson is all about area.

If you're ready to begin, let's go.

The outcome of today's lesson is I can explain how to use multiplication to calculate the area of a rectangle.

Our keywords today, which we're going to do in a my turn, your turn style are my turn array, your turn, my turn length, your turn, and my turn width, your turn.

Have you heard those words before? Let's explore what they mean, shall we.

So items including numbers arranged in columns or rows are called arrays.

And you might have had some experience of exploring times tables, for example, using arrays.

So this is an array made of 12 counters.

Length and width are two words used to describe the sides or dimensions of polygons.

Have a look at that example there.

We're going to be exploring that concept today in quite a lot of detail, so don't worry if you're not too familiar with it yet.

Our lesson is split into two cycles, the first will be identifying the dimensions in a rectangle, and the second will be calculate the area of rectangles using multiplication.

But if you are ready to begin, let's start by identifying the dimensions in a rectangle.

In this lesson, you're going to meet Lucas and Sam, you may have met those two before.

They're going to give us a little bit of a helping hand today day.

The dimensions of a rectangle can be labelled.

So we could call this dimension the width, and we could call this dimension the length.

And they meet, you may notice at a vertex, these two sides are connected at a vertex.

And one side is adjacent to the other side.

The dimensions of the rectangle have changed, but they can still be labelled in the same way.

So this time we're gonna call this dimension the width.

So what do you think we're gonna call this other dimension? The length.

So that same rectangle that we just looked at has been rotated.

So it's exactly the same, it's got the same dimensions, the same area, all that kind of stuff.

It's been rotated.

The two sides are still perpendicular and still connected by a vertex.

So the shape still has a length and a width.

Here we go.

Could call that the width.

So we call this side the length.

What do you notice this time? The same rectangle has been rotated.

Do you think it's still got a length and a width? It has.

The two sides are still perpendicular and still connected by a vertex.

So the shape still has a length and a width, and they can be labelled as such.

So if we're calling this the width, what do you think we're going to call the other side that's perpendicular to it? The length.

So width is a distance from side to side as you can see on that example.

But length is also the distance from side to side.

So the words can be used somewhat interchangeably.

Therefore it does not matter which of the two connected adjacent sides is labelled as length.

So in this case, we've got the bottom sides called width, but we could have called that length, so we could have swapped them over.

Like so.

It's still got a width and a length.

Okay, what about this? What do you think of this? Is this the width or the length? It goes from side to side, doesn't it? Hmm.

There is a measurable distance between the two vertices.

However, it's not the length, it's not the width either.

It's simply the diagonal.

Have a look at this one, what do you think? Has that been labelled correctly? Is that the length and that the width? Well, they certainly do go from side to side, don't they? What do you think? The dimensions of this rectangle cannot be labelled as such, that's wrong, because they're not perpendicular or connected to vertex so we can't say that that is the length and the width.

That's a non-example.

Okay, let's have a check, which of the following rectangles have correctly labelled dimensions? Explain why or why not.

We're going to do a few of these.

So look at A and B, pause the video.

Okay.

Are they correct? Yes, we could call that the length and the width for A, what about B? Yes, they're both correct.

Just been swapped over, but they've still got a width and a length that meet at vertices and are perpendicular.

Let's have a go at another one.

Another pair.

Okay, correctly labelled or not? So have a look at C, is that length and width? And what about D? Pause the video.

What did you think to that one? Well, yeah, even though it's been rotated, C still has a length and a width.

They meet at a vertex, and they are perpendicular to each other.

What about D? No.

No, D is not an example, the two sides are not perpendicular and they do not meet at the vertex so we can't say that's the width and the length.

That's either the width and the width or the length of the length, depending on what you label it.

Okay, what about e and f? Pause the video.

What do you think to e and f then? Hmm.

E, is that correctly labelled? No, but f is.

So e is incorrect.

What's been labelled the width is in fact the diagonal.

So they're not perpendicular.

But f is correct, that is the width and the length.

But you might notice that the width this time is the top part of the shape, but they do meet at a vertex and they are perpendicular so that's why that's the width and the length.

Hopefully that makes sense.

Okay, time for some independent practise, so label these rectangles with the word length and width, and see if you can do it differently each time.

So think about as many different places as you can put the words length and width.

And for task two, complete the sentence using the words below, so we've got the words length, perpendicular, vertex, and width.

And our sentence with the gaps in is when two sides are hmm to each other and meet at a hmm.

The sides can be labelled as the hmm and the hmm.

Pause the video.

I'll see you soon for some feedback.

Okay, how did you get on with that? Let's have a look.

There are lots and lots of different ways that you could have done this and you won't have the same as this I'm sure, but here's just some possibilities.

So if you label that the width, you could label that the length.

Here, we could call that the length and we could call that the width.

You might notice on this occasion, the width is on the right hand side of the shape, but they are still meeting it a vertex and they are perpendicular to each other.

So that's why that's the length and the width.

Here, we might have that as the length, so if we're calling that the length, we've got two options for the width and that's the one we've gone for on the left this time, could call that the width.

And then here we could say maybe that that's the width, and maybe use the top side this time as the length.

But once again, they are meeting at a vertex and they are perpendicular, so that's why they're the length and the width.

And the stem sentence completed goes when two sides are perpendicular to each other and meet at a vertex, the sides can be labelled as a length and the width.

Now you might have also said when two sides are perpendicular to each other, and meet at a vertex, the sides can be labelled as the width and the length.

That would make sense too.

Cycle two is calculate the area of rectangles using multiplication.

So let's hope your multiplication skills are on form.

Let's go.

Sam knows that if she counts the number of square centimetres in the rectangle, it will give the area of the rectangle.

Hopefully you know that too.

So we've got centimetre square here.

She uses counters to help her count the squares.

Let's count with her.

Are you ready? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

I think there was a quicker way to do that, don't you? We could also have counted that using the words centimetre square, so one centimetre squared, two centimetre squared, et cetera.

But either way, that's got an area of 12 centimetre squared.

The counters were helpful.

Now the letter A with a capital can be used to mean area and you can do that, that's the shorthand that mathematicians use, so capital A means area, and it just saves you right in the word area each time.

So A or area equals 12 centimetre squared.

Seeing the counters arranged in an array like this reminds Sam of something.

Hmm.

Does it remind you of anything? This is just like times table she says.

Yes, it is, isn't it? This is a three by four array or a three times four array.

So instead of counting in ones, I could count in threes.

Can you see that? Let's have a look at that, we could go, three, six, nine, 12.

12 centimetres squared.

That was certainly quicker than counting them individually, wasn't it? Can you see another way as well? Instead of counting in threes, I could count in four she says.

Yeah, that's the other way to do it.

You ready to do that? So we go four, eight, 12.

12 centimetre squared.

A or area equals 12 centimetre squared.

But really I don't need to do either of those because I know my times tables off by heart.

Three times four equals 12 she says.

Fantastic.

So if you are really good with your times table and you know them off by heart, that's going to save you a lot of time.

Hopefully you do.

Either way, area equals 12 centimetre squared.

All I need to know she says is the length and the width of the rectangle and then I can use my times table's knowledge.

So if the length is three centimetres, we don't need counters for this, the width is four centimetres, that's our length and width.

We can multiply those together, three times four, three by four equals 12, area equals 12 centimetres squared.

No need for counters and no need for counting.

Sam's teacher has covered up part of this rectangular array with a piece of paper.

Is it still possible do you think to work out the area of the rectangle? So even though you can't see all of those squared, can you still work out the area? Is there enough information there do you think? I think there is, think about what we've just been learning about.

Sam thinks it's too, she says, "I know there are six squares in a row." Yep.

And four rows, One, two, three, four.

Six times four equals 24.

And that was an automatic known fact for Sam, and hopefully it is for you too.

So the area is 24 centimetre squared.

And we could count those to prove it, but we don't need to.

Let's do a check.

How many different ways can you think of to work out the area of this rectangle? Choose the most efficient.

Pause the video and give that a go.

Okay, how did you get on with that? Did you manage to find an efficient way to do that? Well, you could count the individual squares, that would work.

It's gonna take for, not forever, it's going to take a long time though, isn't it? I think there's a quicker away, you could count in threes, yes, that's quicker.

So you could go three, six, nine, 12, 15, 18, 21.

That was certainly quicker than counting them individually.

But is there a quicker away still? I think there is counting sevens, seven, 14, 21.

That was quicker, but I think there's an even quicker way still, and that is knowing our times tables facts.

So three times seven or seven times three is 21.

The final option is definitely the quickest and the most efficient, whichever way you choose, the area of the rectangle is 21 centimetres squared.

Sam says, "I don't even need to see the array inside the rectangle as long as I know or can measure the length and the width.

' Let's have a look at that.

So there's no squares this time.

But she knows the length.

So the length's been given, it's three centimetres, the width is four centimetres.

We know that three times four equals 12, so therefore the area is 12 centimetres squared.

So we've got a generalisation.

To find the area of a rectangle, multiply the length by the width.

That's really important.

So I want us to say that together.

Are you ready? To find the area of a rectangle, multiply the length by the width.

And now I want you to say it all by yourself after three, one, two, three.

Fabulous.

So here we've got those two methods side by side, and you can see they give exactly the same answer.

So we don't actually need to see the squares inside the rectangle to calculate it as long as we know the length and the width, we can multiply them together.

Okay, so we've got that generalisation to find the area of a rectangle, multiply the length by the width.

The length is four centimetres, the width is three centimetres.

The numbers in this rectangle are the same as the previous rectangle.

The shape has been rotated.

However, the area has not changed.

It's the same shape as before, just rotated same area.

Four times three equals 12 once again.

The area is 12 centimetres squared.

Let's have a look at this one, so a new example with a new length and width.

So we've got either the length is three centimetres and the width is 11 centimetres, or the width is three centimetres and the length is 11 centimetres, either way.

So we've got our generalisation area equals length times width.

Stem sentences can be used when calculating the area of a rectangle.

So our stem sentence is this, the length is hmm centimetres, the width is hmm centimetres, hmm times hmm equals hmm.

The area of the rectangle is hmm centimetres squared.

Can we read that again? This time do it with me.

Are you ready? The length is hmm centimetres, the width is hmm centimetres.

Hmm times hmm equals hmm.

The area of the rectangle is hmm centimetre squared.

So let's see if we can fill this in, so have a think about it.

So think what the length might be, think what the width might be.

Remember they can be swapped over, and then what the area would be.

Okay.

Well, the length is three centimetres, so let's choose that.

So if I've chosen the length as a three centimetre side, what's the width going to be? The width is 11 centimetres.

What's my calculation going to be? Three times 11 equals, hopefully this is a times tables fact, hopefully it's automatic to you, you don't need to use fingers or anything like that or strategies, it should hopefully come straight to you, is 33.

So the area of the rectangle is 33 centimetre squared.

Let's read that complete sentence together, do it with me this time, ready? The length is three centimetres, the width is 11 centimetres, three times 11 equals 33.

The area of the rectangle is 33 centimetres squared.

Okay, new numbers, new length and width.

Let's have a go, have a think about it.

So if we choose a length, a seven centimetres and we could have chosen eight by the way as well and swapped it over, but let's say we choose seven centimetres, the width is eight centimetres.

So what's our calculation going to be? Seven times eight.

Now times tables fact hopefully, instant recall fact for our hope, seven times eight equals 56.

So the area of the rectangle is 56 centimetres squared.

Shall we read that together? Let's do it, the length is seven centimetres, the width is eight centimetres, seven times eight equals 56, the area of the rectangle is 56 centimetres squared.

So it does not matter which side is regarded as a length and which is regarded as a width, the area is the same either way.

So we could do it the opposite way around, the length is eight centimetres, so what would the width be? Seven centimetres, so we've just swapped those two over.

Eight times seven equals 56.

The area of the rectangle is 56 centimetres squared.

So it's the same either way, whichever you chooses the length and the width.

Okay.

So the same times tables fact is being used in both of these examples.

So in the first one we've got nine times six, and in the second one we've got six times nine.

Now you might know something about what happens when you swap the numbers over in times tables.

We can use the same stem sentence for it.

So we can say the length is nine centimetres, let's say we've chosen that one for the length.

So therefore the width would have to be six centimetres.

So our calculation is nine times six equals, do you think 54.

Well done if you got that straight away.

The area of the rectangle is 54 centimetres squared.

And that stem sentence applies to both of those shapes.

Let's read that sentence again together, the length is nine centimetres, the width is six centimetres, nine times six equals 54, the area of the rectangle is 54 centimetre squared.

Let's have a check for understanding, select all the calculations that will give the area of this rectangle, so it's a six by 10 rectangle.

So we could either say six is the width and 10 is the length, or six is the length and 10 is the width.

Doesn't really matter.

Okay, which ones give the area? Is it A, 10 plus six equals 16? Is it B, 10 plus six plus 10 plus six equals 32 centimetre squared? Is it C, 10 times six equals 60 centimetres squared? Is it D, six times 10 equals 60 centimetres squared? Is it E, 10 times 10 equals a 100 centimetres squared? There's more than one correct answer here.

Pause the video and give it a go.

Did you manage to find the correct answers? Well, let's have a look.

10 times six is 60 centimetres, that one works, multiplying the length and the width gives us the area there.

There's one more that's similar to that, but swapped over, which is D, six times 10 equals 60 centimetres squared.

If you thought it was B, I think you are still thinking about perimeter.

So this is a lesson about area.

Okay, let's do another check, you're going to calculate the area of this rectangle and we've given you the stem sentence there or the generalisation I beg your pardon there.

So we've got a length or width of six centimetres, then a length or width of 12 centimetres.

What's the area of that rectangle? Pause the video.

Well again, I hope that was automatic to you that you didn't need to use any strategies, but if you did, that's fine too, but it's just quicker if you can do that automatically, this is a timestables fact you might recognise, six times 12 equals 72 centimetres squared.

So the area of that rectangle is 72 centimetres squared.

Very well done if you've got that, you are on track.

Let's do some independent practise.

So number one, complete the stem sentences to work out the area of the rectangles.

So the length is hmm centimetres, the width is hmm centimetres, hmm times hmm equals hmm.

The area of the rectangle is hmm centimetres square.

And then the same for B, there's something different about B though, isn't there? Can you spot what's different? Complete the sentence, and the same for C, and once again, a couple of things that are different about C.

But you can still use that stem sentence to work out the area.

And for number two, calculate the area of the following rectangles.

So we've given you the length and width, but this time no stem sentence, let's see if you can still do it.

Remember A stands for area.

In the second example, we've not given the unit, you've got to write the unit in.

And for number three, little bit of thinking to do, calculate the area of this square.

Now you might look at that and think, "Hang on, there's only one dimension there, that can't be right." Hmm.

I think you can still work it out.

Remember it's a square.

Okay, pause the video.

Good luck with that and I'll see you soon for some feedback.

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

So number one then, the length is five centimetres, the width is four centimetres, five times four equals 20.

The area of the rectangle is 20 centimetres squared.

And we could accept it the other way round as well, so instead of five times four, four times five.

For B, the length is five centimetres, the width is four centimetres, five times four equals 20, the area is 20 centimetres squared.

For C, it's six times seven, so the area is 42 centimetres squared.

And again you might have given them in a different order, that's fine.

For number two area is 48 centimetre squared because eight times six is 48.

And for the other one, it is 84 centimetre squared, seven times 12, one of our known times tables facts hopefully is seven times 12 is 84 centimetres squared.

And for number three, the area of the square is 36 centimetres squared 'cause if it's a square and the length is six centimetres, that means the width is also six centimetres.

All the other way around if you want to look at it that way, 'cause it's a square, so the sides are the same length.

So six times six and known times tables fact I hope is 36 centimetres squared.

We've come to the end of the lesson.

Today's lesson has been explaining how to calculate the area of a rectangle using multiplication.

Our times tables facts were really important there, weren't they? So all rectangles have a length and a width.

And remember it doesn't matter which one we label as the length and the width, as long as they meet at the vertex and they all perpendicular to each other.

To find the area of a rectangle, multiply the length by the width.

We've got that generalisation there, look area equals length times width.

So in this example, three times 11 equals 33.

So the area of that rectangle is 33 centimetres squared.

Well, I hope you've had fun in today's lesson, I definitely have.

Hope you've learned lots in today's lesson, some really important knowledge today.

It's been a great pleasure working with you, and I really hope to see you again soon for some more maths.

But until then, take care of yourselves and goodbye.