Lesson video

Hello, I'm Mr. Tilston.

I'm a teacher.

If I've met you before, it's lovely to see you again.

And if I haven't met you before, it's lovely to meet you.

Hope you're having a great day and that you've been very successful so far today.

Let's make it an even better day and an even more successful day by tackling this lesson, which is all about perimeter and solving problems to do with perimeter.

So if you're ready, let's begin.

The outcome of today's lesson is this.

I can describe the relationship between scale factors and side lengths and perimeters of two shapes.

And our keywords are, and some of these will be familiar to you and some won't.

My turn, perimeter, your turn.

My turn, scale factor, your turn.

And my turn, similar, your turn.

Okay.

Two of those words I think you are very confident with.

Let's remind you what they mean.

And then let's talk about scale factors.

Perimeter is the distance around a 2D shape.

You knew that, didn't you? A scale factor is the multiplier between similar shapes that describes how large one shape is compared to the other.

Now that will make sense soon, I promise.

Two shapes are similar when one shape can become the other after a resize, flip, side or turn.

So we're looking for similarities in shapes and that's just some of the ways that we can do it.

Okay, our lesson is split into two cycles today.

Two parts.

The first will be scale factors, side lengths and perimeters.

And the second, the link between scale factors and areas.

So if you are ready, let's begin by thinking about scale factors, side lengths and perimeters.

Let's learn about scale factors.

In this lesson, you're going to meet Alex, have you met him before? He's here today to give us a helping hand with the maths.

Look at these pictures of Alex.

What's the same and what's different about these photographs of Alex? Have a good look.

Take your time.

Good mathematicians notice things.

What do you notice? Hmm? What do you notice about A? What do you notice about B? What's the same and what's different? Let's see.

Well, they're both rectangles.

Did you say that? Well done if you did, that's something that is the same about them.

The rectangles are very similar, but the side lengths are different.

Did you notice that? Well done if you did.

They are twice as long in photograph B as photograph A.

Did you notice that? So let's have a look.

So in photograph B, the short side is eight.

And in photograph A, the short side is four.

So yes, eight is twice four.

And in photograph B, the long side is 12.

And in photograph a six.

So yes, 12 is twice six.

So they're half as long in A as in B.

So we could look at it the other way round.

The length and width have both doubled.

Scale factor can be used to explain the relationship.

So this is where we're going to start to think about that terminology scale factor.

And we've got a stem sentence here and the stem sentences as follows, to change shape mm into shape mm, scale the side length by a scale factor of mm.

Let's explore that and let's start to fill in some gaps.

So this case we've got shapes A and B.

So that's what we're going to say here.

To change Shape A into shape B, scale the side lengths by a scale factor of, now they doubled, didn't they? So what number could we use here do you think? Two.

So they've doubled, they've been multiplied by two.

The side lengths have been scaled by a scale factor of two is how we can describe that.

The scale factor describes the multiplicative relationship.

When the scale factor is greater than one, the shape has been made larger, and in this case it is greater than one.

It's two.

So that means the shape's going to be larger.

The scale factor's not always greater than one, sometimes it's less than one.

And in which case the shape becomes smaller.

Scale factor can be used to describe the transformation from B to A.

So let's think about it the other way round.

Same shapes, same dimensions, same stem sentence to change shape mm into shape mm.

Scale the side length by a scale factor of mm.

Okay, let's explore this.

How could we fill this in this time? Well, to change shape B into shape A, scale the side length by a scale factor of what? Can't be two, can it? 'cause that would make it bigger, but this has been made smaller.

It's been made half the size to be specific.

And that's what we say, a scale factor of one, half.

The side lengths are one half times the size of B.

If the scale factor is less than one, the shape has been made smaller.

So we've seen examples now where the scale factor's greater than one and the shape's bigger.

And we've seen examples where this scale factor is less than one and the shape has been made smaller.

Now what could be said about the perimeters of the shapes? What do you think? Would you like to have a look and work it out? What are the perimeters of the shapes? Is there a link? Well the perimeter of this shape is 20 centimetres and I'd work that one out by doing six ad four and doubling it.

And what about the perimeter of B? Well using that same method, 12 plus eight is 20, doubled is 40.

So that's got the perimeter of 40 centimetres.

Now what do you notice? Is there a link between those two perimeters? Because each side length from shape A has doubled.

The perimeter has also doubled.

Well that makes sense, isn't it? To change shape A into shape B, scale the perimeter by a scale factor of what? So you know about scale factors.

This time the perimeter's doubled.

So it's two.

And what about the other way round? Change shape B into shape A, scale the perimeter by a scale factor of what? It's halved.

So one half.

Rectangles A and B are what we call similar.

Similar shapes are shapes where the sides have been enlarged by the same scale factor.

The interior angles remain the same.

Scale factors can be used for other shapes.

So we've seen it with a rectangle.

Let's have a look at triangles now.

Now once again, the triangles are similar, the angles are the same, for example.

Let's have a look at this stem sentence again to change shape mm into shape mm.

Scale the side length by a scale factor of mm.

What do we think? Wow, let's go from C to D.

To change shape C into shape D, scale the side lengths by a scale factor of what? What's the relationship between these shapes? What's the multiplicative relationship? How many times do you multiply three by to get six? Two.

And the same with multiplying two by two to get four.

What about the other way around? Let's go from D to C.

To change shape D into shape C, scale the side lengths by a scale factor of what? What's the opposite of multiplying by two? A scale factor of one half.

When a shape has been transformed by a scale factor, the perimeter is also transformed by the same scale factor.

So let's look at this slightly tweak stem sentence.

To change shape C into shape D, scale the perimeter by a scale factor of two.

And to change shape D into shape C, scale the perimeter by a scale factor of one half.

I think you are getting this.

So there we go.

The perimeter's eight and the perimeter here is 16.

Well all of our examples so far have involved scale factors of two and one half, but other scale factors do exist.

So what do you notice about these two similar rectangles? What could we say? And we've got our stem sentences again, the change shape mm into shape mm, scale the side length by a scale factor of mm.

Let's see if we can do that both ways round.

Let's start by changing shape E into shape F.

Let's look at that relationship.

Let's take the long side.

The long side of the small shape is three centimetres and the long side of the bigger shape is nine centimetres.

What's the multiplicative relationship between three and nine? How many times do we multiply three to get nine? Three.

So this time the scale factor is three and then going the other way round.

So to change shape F into shape E, scale the side length by a scale factor of what? Now we're making the shape smaller.

So we need a number less than one.

Before we used a fraction, didn't we? We used half.

What fraction could we use now? If the scale factor was three to go from E to F, what would it be to go from F to E? Is three a clue? Yes.

One third.

So to change shape F into shape E, the scale factor is one third.

Let's have a check.

We've got those stem sentences.

Can you fill those in? This time it's about perimeter.

Pause the video.

Well hopefully you worked out that we've already done the hard work here when we worked out the scale factor, because if we're scaling the side lengths by a certain scale factor, the perimeter's going to be scaled by the same scale factor.

So again, it's three to change shape E into shape F, scale the perimeter by a scale factor of three and then to change shape F into shape E, scale the perimeter by a scale factor of what fraction? One third.

Very well done.

If you've got those you are on track.

The perimeter's 10 centimetres here and 30 centimetres here.

So 30 is 10 multiplied by three.

That's the scale factor.

Let's have a look at these examples, are rectangles D and C similar would you say? What do you think? Would you say they're similar? Well the short side of D is slightly more than double the short side of C.

Let's check that out.

So double eight 16.

But that side is 17, so it's a bit more than double.

And the long side of D is slightly less than double the short side of C.

So yes, double 12 would be 24, but that's 23.

So it's a little bit less.

So can we say they're similar? Have they been scaled by a scale factor? The scale factor is not consistent.

The rectangles are not similar despite what your instinct might have been.

'cause they've not been scaled by a scale factor.

They're not mathematically similar.

Okay, have a look at this example.

There's something a little bit different.

Can you be eagle eyed and spot what's different this time? It's something different that's known and something different that's unknown.

While this time we actually know the scale factor to change shape G into shape H scale the perimeter by a scale factor of two and to change shape H into shape G scale the perimeter by a scale factor of one half.

So the scale factor's known what is not known.

Can you see? It's the side lengths of H, but can we work it out now that we know the scale factor? Yes we can.

We can use multiplication.

Multiply the side lengths of G by two.

That's the scale factor.

Six multiplied by two.

Nice, easy question for you.

There is 12 centimetres.

And what about the other side? The longer side, what's that going to be? Or seven multiplied by two.

Nice and easy.

14.

So that is the side length of H.

Okay, let's pause and have a little check.

Can you fill in the blanks to change shape I to shape J? Scale the side lengths by a scale factor of what? And then the other way round, please.

Pause the video and have a go.

Right, what did you come up with here? Let's have a look.

Four.

That's a multiplicative relationship.

We can see if I has got a long side of three and J's got a long side of 12, you can say three times four is 12.

So that's a scale factor.

And then to change shape J into shape I, scale the side lengths by a scale factor of what? What would the fraction be? One quarter.

Very, very well done if you've got that.

I think it's time for some practise.

Let's see what you've learned about scale factors.

Number one, identify the scale factor for each pair of shapes.

And we've got that stem sentence that hopefully you're getting very familiar with now to help you out.

So see what you can notice.

And the same for B.

And again the stem sentence are there to help.

It's a different scale factor this time.

And the same for C.

You might notice something a bit different this time about the shapes and the position of the shapes.

Can you spot it? And number two, shape G has been scaled by a scale factor of three.

So we know the scale factor this time to make shape H, right? The missing side lengths on shape H.

And then when you've done that, find the perimeter of both shapes.

And I wonder if there's a quick, easy way to do that.

I think there is.

Number three, if the side lengths of these regular shapes are scaled by a scale factor of three, what will the perimeters of the new shapes be? Okay, think about that's a scale factor of three.

Number four, if the side lengths of these shapes are scaled by a scale factor of one half, what will the perimeters of the new shapes be? Hmm.

And number five, complete the table, which is all about squares.

So let's look at the columns.

The first one is the side lengths of shape A.

Some of those are given, some aren't.

The second one is the scale factor.

Again, some are given, some aren't.

The third one is the side lengths of shape B.

The fourth one is the perimeter of shape A and the fifth one is the perimeter of shape B.

So you can work out hopefully the missing information by using the known information.

You might have to think about that, but it's all there, right? Well good luck with that and I'll see you soon for some feedback.

Welcome back.

How did you get on with that? Here we go.

So number one, the scale factor.

Well two multiply by two equals four.

So that's the scale factor.

The scale factor is two.

What about the other way round to go from B to A? That's one half.

And for B, what's a multiplicative relationship between six and 36 times five is 30.

So that's a scale factor.

And then to go the other way round from D to C, that's one fifth, that's a scale factor.

For C, let's have a think about C.

Now to go from E to F, we're making the shape smaller, so we need a scale factor that's less than one.

Hmm.

How many times smaller though? How many times does eight fit into 24? Three.

So the scale factor is the fraction, one third.

And then the other way round to go from F to E.

It's a scale factor of three.

And then shape G's been scaled by a scale factor of three.

So we're multiplying those side lengths by three.

12 times by three is 36 and four multiplied by three is 12.

So they are the new side lengths.

Just times tables, isn't it? The perimeter is 32 centimetres and the perimeter of H is 96 centimetres.

They've got the same scale factor of three.

Number three, if the side lengths of these regular shapes are scaled by a scale factor of three, what will the perimeters of the new shapes be? So let's have a look.

Well the perimeter of this shape is 20 centimetres and 20 multiplied by three equals 60.

So 60 centimetres.

And then what about B? Well, remember it's a scale factor of three.

The perimeter of that would be 24 multiplied by three equals 72 centimetres.

And then for number four, the scale factor is one half.

So we're making these shapes smaller.

And then thinking about the perimeter, so the perimeter's going to be half two.

So 28 centimetres is the original perimeter divided by two equals 14 centimetres.

And the same here, 24 centimetres divided by two equals 12 centimetres.

Number five, complete the table.

It's all about squares.

So the first one is 60 centimetres, that's a perimeter.

And then we've got three centimetres, 48 centimetres and 12 centimetres, 10, 16 centimetres and 160 centimetres, three kilometres, 12 kilometres, 12 kilometres, 48 kilometres, and then 21 millimetres, seven millimetres, 84 millimetres and 28 millimetres.

And well done if you spotted the changing units there from centimetres to millimetres and kilometres.

You're doing really, really well.

And I think you are now ready for cycle B, which is the link between scale factors and area.

So you know about scale factors now, but let's think about area.

Alex says, "I think I'm getting this now." Yeah, good on you Alex.

A scale factor of two has been used to transform shape A into shape B.

Would you agree? Let's have a look.

Well I can see the perimeter's doubled straight away.

So yeah, I can see the side lengths have doubled.

So yes, it's definitely a scale factor of two.

The side lengths doubled the perimeter, double the area must double two things, Alex.

Would you agree with him? It seems to make sense, doesn't it? It seems to follow the pattern.

I wonder if he's right, we need to investigate this.

Do you agree with Alex? Hmm? You might want to pause and discuss this.

Well the shapes are still similar.

They've got a scale factor.

The area of this shape is 24 centimetre square and we can get that by multiplying those two side lengths.

What about the area of the neck shape B, 12 multiplied by eight.

Well let's have a think about this.

That is a side shape B, that's 24 centimetres.

And there it is again.

And there it is again.

And there it is again.

12 multiplied by eight is 96.

And when you add those areas up, it also equals 96.

Would you say the area is doubled? No.

Shape B can be composed of four copies of shape A hasn't doubled.

So if it's got a scale factor of two, the side lengths are multiplied by two, the perimeter is multiplied by two, but the area has actually been multiplied by four, so he wasn't correct.

But that doesn't change the original scale factor.

It's still two.

The area can be calculated by multiplying the length and the width.

And I know you knew that.

So here we go.

So six multiplied by three is 18 centimetre squared and then 12 multiplied by six is 72 centimetres squared.

So the factors have actually doubled.

How does that affect the product if both the factors have doubled? Well, let's have a look.

So six multiplied by three, six doubled equals 12, three doubled equals six.

So that becomes 12 multiplied by six.

If both factors double, the product will be four times the size.

And we saw that before with the rectangles.

Alex has got a new theory, he's doing what good mathematicians do, he is conjecturing, he says, "I think that if the scale factor is three, you need to multiply the area of the original shape by six to find the area of the new shape." Can you see where he's got that logic from? Why am I to be thinking this? Well in the that we just looked at the scale factor was two and the area was multiplied by four.

So now he's thinking of scale factors three.

You multiply the area by six.

I see the logic.

What do you think? Do you agree? So he's following that logic by saying if the scale factors four, you need to multiply the area by eight.

I see where he is going.

So if the scale factor was five, according to Alex, you would multiply the area by 10.

Yeah.

Okay.

Do you agree with Alex's hypothesis? And you may want to take some time to ponder this before we carry on, but let's investigate.

So Alex says, I think if the scale factor is three, you need to multiply the area by six.

Let's investigate.

So here we've got an example of where the scale factor is three.

You can see that with these side lengths.

So three multiplied by three is nine, two multiplied by three equals six.

So it's got the scale factor of three.

But let's think about the area, two multiplied by three and six multiplied by nine.

Two multiplied by three equals six.

So the area of the small rectangle is six centimetres squared.

Nine multiplied by six equals 54.

So the area is 54 centimetres squared.

Now Alex said that you need to multiply the area by six, but six times six equals 36.

And that's not the area of the bigger shape, it's 54.

Hmm.

So how many times do you multiply six by to get 54? Nine.

So in this example the scale factor was three, but the area of the second shape was nine times greater than the first.

Hmm.

And his hypothesis was if the scale factor is four, you need to multiply the area by eight.

Well what do you think? Let's have a look.

Here's an example of a scale factor of four.

Think about the areas.

Well the area of this small shape is four centimetres squared and the area of the larger shape, well let's think about that.

How many times can you fit in that small shape into the larger shape? It's not eight is it? It's more than eight.

It is truer to say that squaring the scale factor tells you how many times to multiply the area when transforming the shapes.

So in this case, the area of the small shape was four centimetres squared and four squared is 16.

So we multiply the area by 16 in this case to get the area of the new shape.

So when the scale factor's four, you multiply the area by 16.

Scale factor four, four squared equals 16, four centimetres squared multiplied by 16 equals 64 centimetre squared.

That's the area of the larger parallelogram or rhombus.

Let's do a quick check for understanding.

If shape A has been scaled by a scale factor of three, what happens to its area? It's multiplied by three, it's multiplied by four, it's multiplied by nine or the area does not change.

What do you think? Discuss that with a partner if you can and pause the video.

Okay, what do you think if shape A has been scaled by a scale factor of three, what happens to its area or three squared is nine, so it's multiplied by nine.

So well done if you've got that.

It's time for some final practise.

Number one, if a shape is scaled by a scale factor of five, what happens to the area of the original shape? So we left off there by noticing that when the scale factor was four, we multiply the area by 16.

So what, when it's five? Number two, if this shape is scaled by a scale factor of three, what is the area of the new shape, the area of the new shape? And number three, if the same shape is scaled by a scale factor of one half, what's the area of the new shape? And number four, complete the table, which is all about squares.

And again, some information is given and some is missing, but you've got enough there to work out the missing information.

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

Welcome back.

How did you get on? Number one, if a shape is scaled by a scale factor of five, what happens to the area of the original shape? Well we need to square it, don't we? Five multiply by five equals 25.

So the area will be 25 times the size of the original.

And if the shape is scaled by a scale factor of three, what's the area of the new shape? Well the area of the original shape is multiplied by nine and the area of the original shape is 32 centimetres squared.

So that's 32 multiplied by nine and that's 288 centimetres squared.

You may have needed a written method for that I think I would've done.

And number three, if the same shape is scaled by a scale factor of one half, what's the area of the new shape? So it's still got an area of 32 centimetres squared, but scaled by one half, a bit trickier.

Well this time we divide it by four.

So 32 divided by four equals eight centimetres squared.

Well then if you got that one, that was a bit trickier.

And number four, complete the table.

Here's the missing area of shape B, 225 centimetre squared and we've got shape A, 144 centimetre squared, shape B, nine centimetre squared, shape A, 16 centimetre squared and shape B, 1,600 centimetres squared.

We've come to the end of the lesson.

Today, you've been describing the relationship between scale factors and side lengths perimeters of two shapes.

And I'll bet at the start of the lesson you didn't really know anything about scale factors, but I bet you know lots and lots now.

When a shape is transformed by a scale factor, the side lengths and the perimeter are transformed by the same scale factor.

Now in this example that we can see here, the scale factor from A to B is two, and the scale factor from B to A, the other way round is half.

So six multiplied by two is 12, and half of 12 is six and four multiplied by two is eight, and half of eight is four.

So it's all about that multiplicative relationship.

You've been amazing today, you've learned some new things and you've done really, really well with it.

So give yourself a pat on the back.

It's very well deserved.

I hope you have a great day, whatever you are doing next.

And until the next time, take care and goodbye.