Lesson video

Hello and welcome to this lesson on ratio and proportion.

By the end of the lesson, you should be able to identify and describe the relationship between regular polygons using scale factors.

Lots of language in here today.

So let's have a look at that language and how we're going to be using it in this lesson.

So the key words we're going to be using in this lesson are dimensions, scale factor, regular polygon, and similar shapes.

So some of those will be new, I expect.

So you might want to have a pause at this point.

Pause the video, have a think about what you know about these words, or think you might know.

So let's think about what these words mean and how we are going to be using them in this lesson.

So the dimensions of an object or a shape are the lengths of its sides and edges and that's gonna be really important in what we're thinking about today.

Thinking about the dimensions of shapes, the lengths of the sides and the edges.

Now, a scale factor is linked to ratio.

So a scale factor is the ratio between the dimensions of the original shape and a new shape.

And we know that factors are linked to multiplication.

So we're going to be multiplying up, or multiplying and dividing the dimensions of shapes in order to create new shapes.

The shapes we are looking at in this lesson are regular polygons.

So a polygon is a shape with many sides, and those sides are all the same lengths and the angles are all equal.

So for a regular polygon, the sides are all the same length and the angles are all equal.

We'll be looking at regular polygons in this lesson.

Similar shapes are what we are going to be creating.

So similar shapes can be transformed into each other, or changed into each other by scaling their dimensions by the same scale factor.

Now, that's all going to become much clearer as we go through the lesson.

So we've brought together a lot of our key words into that final definition for similar shapes.

So similar shapes are all about creating a new shape by using a scale factor and by changing, transforming the dimensions of an original shape into a new shape.

So watch out for those words and those definitions as we go through this lesson.

So in the first part of the lesson, we're going to be comparing regular polygons.

Just looking at two regular polygons and seeing what's the same and what's different? And gradually, we'll make that more and more mathematical.

We'll come up with an equation.

Let's have a look at some shapes.

I've got two shapes here, shapes A and C.

So what can you say about shapes A and C? You might want to pause the video here and have a think a think about what you would say, and then we'll look at some statements about these shapes.

So I've got some statements I'm going to bring up here and I'm going to see if you agree with these statements.

So it says here, shape C is bigger than shape A.

Well, I think we'd all agree that shape C is bigger than shape A, so we'll give that one a tick.

Shape A is smaller than shape C.

Well, yes, that's the opposite of what we've just said, isn't it? Shape A is smaller than shape C so we'll give that a tick as well.

Shapes A and C are both squares.

Well, yes, A square has all its sides the same length.

So A is a square and C is a square.

So, yes, I'm happy with that as well.

Shape C is three times as big as shape A.

Hmm, that's an interesting one.

You might wanna pause and have a think about that.

Is shape C three times as big as shape A? I'm not sure.

I think we need to investigate that a bit further.

So, what is meant by three times as big? What are we talking about here? So let's explore and see what we think.

So let's remember that word dimensions is one of our key words today and that when we talked about similar shapes and scale factors, we talked about them relating to the dimensions of a shape.

So when we are talking about scaling shapes, making shapes bigger, at the moment, certainly in this lesson, we are talking about making the dimensions bigger.

So let's have a look at A and C and their dimensions.

So the side lengths of square A are one, one unit on our squared paper.

The side lengths of square C are three units long.

So we can say that the side lengths, the dimensions of square C are three times the side lengths of square A.

So the side length of square A multiplied by three is equal to the side lengths of square B, well, multiplied by three is equal to three.

So in this sense, we can say that shape C, square C is three times the size of square A.

Its dimensions are three times the size of the dimensions of square A.

Another key word there was similar shapes.

So can we say that A and C are similar shapes? Well, the dimensions of shape C are three times the dimensions of shape A, and our definition of similar shapes was that they can be transformed into each other by scaling their dimensions by the same scale factor.

Scaling is about multiplying up, isn't it? So, yes, A and C are similar shapes, because we've multiplied all the sides of A by three to transform it, to change it into shape C.

Okay, we've introduced a lot of language, a lot of vocabulary in the beginning of this lesson.

So let's pause and have a bit of a check.

Can we apply some of that vocabulary, that language to these two shapes? So what is the relationship between these equilateral triangles? Are equilateral triangles, regular polygons? Well, let's see, the definition was that all the sides are the same length and all the angles were equal.

And an equilateral triangle has all sides the same length.

So all the angles are equal.

So, yes, they are regular polygons.

So you might want to pause the video here and have a think about what you feel the relationship is between these equilateral triangles A and B.

Okay, so let's have a look at their side lengths, their dimensions.

So the length of the sides of A is three centimetres.

The length of the sides of triangle B is six centimetres.

So, what's the relationship there? So we can say that the side lengths of B are two times the side lengths of A.

the dimensions have been multiplied by two, they've been doubled.

So the side lengths of A multiplied by two are equal to the side lengths of B.

Three multiplied by two is equal to six.

So what do you notice about the angles in the triangles? So if you remember, we talked about the fact that dimensions were really important in this lesson.

Dimensions are the lengths of the sides of the shape, the sides of the edges.

The angles are going to stay the same, because what we've done is create similar shapes here.

Similar shapes have to be the same basic shape, and they're similar because their dimensions have been multiplied or scaled by the same factor.

So A and B are similar shapes, their side lengths have all been multiplied by two.

The side lengths of A have been multiplied by two to create shape B, to transform it into shape B, but the angles stay the same because they are still both equilateral triangles, and they have three angles of 60 degrees.

Time for you to have a practise, and a lot of it is a practise around all that new language we've been using.

So you are asked to explain how shape D has been transformed into shape E, and are shapes D and E similar? And explain how you know.

Task two asks you, in a, to explain how shape B has been transformed into shape C.

B asks you to draw another square, square D, which has sides 1 1/3 times the size of C.

And part C asks, how is square D related to square B, so the one you've created and how is it related to square B? So pause the video, have a go at your two tasks, and then we'll look at the answers together.

So task one was looking at shapes D and E, which I'm sure you realise were hexagons, regular hexagons because all the sides are the same length.

So what has happened to transform shape D into shape E? So what can we say? So the lengths of all the sides of E are three times the size of the sides of D.

The length of the size, the dimensions of hexagon D, each side is two centimetres long.

Each side of hexagon E is six centimetres long, which is three times the size.

So are shapes D and E similar? Yes, they are, they're both regular hexagons and all the sides of D have been multiplied by three to create shape E.

So all the sides of E are three times the length of D.

So therefore these are similar shapes, they're both regular hexagons and they are linked by that fact that we've multiplied all the sides by the same factor.

Task two, now this was interesting, 'cause we've got some fractions involved here, haven't we? So we need to explain how shape B has been transformed into shape C.

So the length of the sides of B is two, and the length of the sides of square C are three.

So what can we look at? How can we work out what's happened here? Because we've not neatly doubled anything or halved anything.

I wonder if we can draw B next to C and see what's happened.

So if I put two shape Bs, two of the square Bs onto the dimension of C, we can see that the dimensions of square C are 1 1 /2 times the length of the sides of B.

So the dimensions of square C are 1 1/2 times the dimensions of B.

So now we're asked to draw another square, square D, which has sides 1 1/3 times the size of C.

So you might, if you didn't do it this way, you might want to pause and have a think about doing that drawing again, drawing square C.

And let's have a look and see how we get on.

So here is square C and we want a square which has size 1 1/3 times the size of C.

So 1 1/3 would give us a dimension of four.

1 1/3 times C would give us a dimension of four.

So D will be a square with dimensions of four units.

Part C said how is square D related to square B? So let's have a look.

There are B and D next to each other, and we can see that the sides of square D are two times the length of the sides of square B.

So the dimensions of B have been doubled to transform it into square D.

Interesting that, because C, the square we used, had a fractional relationship to both square B and to square D, but squares B and D have a doubling relationship.

The dimensions have been doubled to transform B into D.

Okay, so first part of the lesson, we've been comparing regular polygons and we've been thinking about that language that we've been using as well.

And the language, one of the phrases that we've used has come up, I've mentioned it one or two times, but we've not not focused on it.

That's the focus of the second part of our lesson, that idea of identifying a scale factor.

So let's have a look at those shapes again and think about them with a scale factor involved.

Okay, so what is the scale factor? So we said that a scale factor was a number that we multiplied all the dimensions by in order to create a similar shape.

So to transform shape A into shape C, we talked about the fact that we'd multiplied by three.

So let's see what that looks like when we're thinking about the language of scale factor.

So we said that side length of square A multiplied by three equaled the side length of square B, because one, the length of the side in A, multiplied by three is equal to three, which is the length of the sides of C.

So all the dimensions of square A have been multiplied by three to make square C.

The dimensions, remember, were the sides, the edges of the shape.

So what we can say is that the scale factor between square A and square C is three.

All the dimensions have been multiplied by three, by that scale factor of three.

So thinking about that practise card task you did at the end of the first part of the lesson, what is the scale factor between square B and square C? So we knew that we'd had 1 1/2 times the side length of B gave us the side length of C.

So the side length of B multiplied by 1 1/2 equaled the side length of square C.

So the scale factor between square B and square C is 1 1/2.

All the dimensions of square B have been multiplied by 1 1/2 to make the dimensions of square C.

So time to have a check as to our understanding of all that language we've been using and this idea of scale factors.

So here are our two equilateral triangles, A and B.

So what is the scale factor between equilateral triangles A and B? You might want to pause the video and have a think, and then we'll talk through it together.

Okay, so we can see that the side lengths of A multiplied by two equal the side lengths of B.

So side length A is three centimetres, three multiplied by two is six, and that gives us the side lengths of triangle B.

So we can say that the scale factor between triangle A and triangle B is two.

The dimensions of A, the length of its sides, have been multiplied by two to give us the dimensions of triangle B, the lengths of its sides.

And there we are, the dimensions of triangle A have been multiplied by two to make triangle B.

We've transformed triangle A by multiplying its side lengths by two, by a scale factor of two, we've transformed it into triangle B.

They're both equilateral triangles, so the angles stay the same.

It doesn't matter how big or small an equilateral triangle's side lengths are, the angles will always be 60 degrees.

Another one just to check where we are with our understanding around identifying scale factors.

So what's the scale factor between squares A and E? You may want to pause the video and have a think, and then we'll look at it together.

Well, we can see here that the side lengths of A are equal to the side lengths of B.

the dimensions of those squares haven't changed.

So the scale factor between square A and square E is one.

The dimensions of square A have been multiplied by one to make square E.

And when we multiply by one, our value doesn't change.

Our product is the same as the factor that has been multiplied by one.

So we can have a scale factor of one, and in fact, that's quite important.

If the shapes are congruent, then the scale factor is one.

Congruent shapes are shapes that are identical in their dimensions.

Sometimes they might be in a different orientation, but for a square, that doesn't really matter.

But congruent shapes have a scale factor of one.

Time for for you to do some practise.

So, task one, what is the scale factor between hexagon's D and E.

And b, what can you say about the angles in hexagon's D and E? So revisiting those hexagons, but thinking about the scale factor and thinking about what happens to angles when we transform shapes using a scale factor.

Task two asks you to identify the scale factor between these pairs of squares.

So these are the squares that we looked at in part one.

So what's the scale factor between squares B and C? What's the scale factor between squares C and D? And what is the scale factor between squares B and D? Now there's a task three this time.

Alex has made a statement here.

Alex says, "Shape P has been scaled by a scale factor of four to make shape Q." Do you agree or disagree with Alex? And explain your answer.

So you may want to pause the video here and have a go at those practise questions, and then we'll look at the answers together.

So in task one, a asked you to work out what the scale factor was between hexagons D and E.

So we can see here that hexagon D, the dimensions of D were two centimetres, the length of the sides.

The lengths of the sides of hexagon E are six centimetres.

So the dimensions of D have been multiplied by three to create the dimensions of E.

So the scale factor between hexagons D and E is three.

Part B asked you what you could say about the angles in hexagons D and E.

And we can see that because they are both regular hexagons, the angles will be the same.

When we transform a shape, when we scale a shape up, we scale its dimensions, the length of its sides and edges.

We don't change the angles, because the angles stay the same because we are creating similar shapes.

And those similar shapes are both regular hexagons, so therefore their angles, their angle properties are the same.

So task two asked us to identify scale factors between different pairs of these squares.

So the scale factor between squares B and C is 1 1/2.

If you look at square B and look at square C, the length of the side of square B is two squares.

Square C is three squares.

So that's one whole lot of B and another half of B.

So 1 1/2 times B.

So the scale factor between squares B and C is 1 1/2.

What about the scale factor between squares C and D? We've got that it's 1 1/3.

How can we we think about that? Well, if we put square C alongside square D, we can see that we've got a whole length of square C in the dimensions of square D plus another one square, one more third of C.

So the scale factor between square C and D is 1 1/3.

And our last pair was squares B and D.

What's the scale factor there? Well, the length of the side of B is two squares, and the length of a side of D is four squares.

So that's been doubled, multiplied by two.

So the scale factor between squares B and D is two.

So task three asks you to think about Alex's statement that shape P has been scaled by a scale factor of four to make shape Q.

Do you agree or disagree, and can you explain why? Well, I hope you realise that Alex is not correct.

Scaling creates similar shapes, and similar shapes are the same shape, but here we can see that shape P is an equilateral triangle and shape Q is a square.

Yes, two centimetres multiplied by four is eight centimetres.

But when we scale shapes, we transform them into similar shapes, and similar shapes have to be the same basic shape.

So we can't use a scale factor to transform a triangle into a square.

So Alex is not correct.

Thank you for your hard work in this lesson.

I hope you've started to understand how we can describe a relationship between regular polygons using scale factors.

We looked at a lot of language in here.

We talked about the fact that when we are scaling a shape, transforming shapes to make similar shapes, we are thinking about their dimensions, the length of their sides, and multiplying them all by the same factor, a scale factor, to transform those shapes and to create a shape with the same properties, a similar shape.

So thank you very much for your work and I look forward to seeing you again.

Thank you.