Lesson video

<v Instructor>Hello and welcome to this lesson on ratio and proportion.

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

So let's have a look at what that all means.

So a key word in this lesson is irregular polygon.

Now you might have come across polygons and regular polygons, you might even have come across irregular polygons.

So what does this little phrase actually mean? Let's have a look.

So an irregular polygon is a polygon shaped with many sides where the sides and angles are not all equal.

So this is any shape, really, any closed shape, made from straight sides.

But some of these irregular polygons are shapes that look quite familiar to us, so it's worth just looking.

So a right angled triangle is an irregular polygon, its sides and angles are not all the same.

Same with an isosceles triangle.

Two sides are the same length, but the third is different, so its sides are not all equal, its angles are not all equal.

A rectangle, as long as it's not a square, which is a special type of rectangle, a rectangle or an oblong is not a regular polygon, it is irregular because it has two long sides and two short sides.

It may have all angles the same, but the sides have to be all the same length too, and they are not in a rectangle, which is not a square.

And finally on the screen there we've got a trapezium, quite a familiar looking shape, but it is an irregular polygon because the sides are not all the same length and the angles are not all equal.

So the shapes we're going to be looking at in this lesson are all irregular polygons.

So there are three sections to this lesson.

In the first section we're going to identify the scale factor in irregular polygons.

So here are two rectangles, A and B.

What can you say about shapes A and B? You might want to pause the video and think about what you can say about shapes A and B.

I wonder what you said about the shapes.

We can see that they are different sizes, but I wonder if you thought about a scale factor that links rectangle A with rectangle B.

Well, let's look at the heights of our rectangles.

The height of rectangle A is one unit, one square.

The height of rectangle B is three squares or three units.

So we can say that the height of B is three times the height of A.

Now that's okay, that's one side, but what about the other side of the shape? Because our shapes do not have equal length sides.

So let's look at the width as well.

So the width of rectangle A is two units, two squares.

The width of B is six units.

So we can say that the width of B is three times the width of A, or A multiplied by three is equal to the width of B, two multiplied by three is equal to six.

So all the dimensions of B are three times the size of the dimensions of A.

So we can say that the scale factor from A to B is three.

We've multiplied all the dimensions of A by three to transform it into shape B, which has dimensions three times bigger.

So the height is three times bigger and the width is three times bigger.

Because the length of the sides are not the same, we have to make sure that the same scale factor has been applied to all dimensions if the shapes are to be similar and if they have been scaled.

Now let's look back the other way because a scale number can be a whole number or it can be a fraction.

So let's think about what's happened to shape B to transform it into A.

So we can say that dimensions of A are 1/3 times the size of the dimensions of B, they've been divided by three, which is the same as multiplying by 1/3.

So the scale factor from B to A is 1/3, three multiplied by 1/3 is one, so the height of those two rectangles.

And if we think about the width, the six which is the width of B multiplied by 1/3 1/3 of six is equal to two, which gives us the width of rectangle A.

So depending on which shape we're starting with and which shape we are transforming to, our scale factor could be a whole number or it could be a fraction.

Let's have a bit of a check here.

So we've got two right angle triangles here, P and Q, and we need to complete these sentences.

So the height of Q is hmm, times the height of P.

The width of Q, we'll consider how that relates to the width of P.

Can we find a scale factor and can we use that scale factor to talk about how the dimensions of those shapes have been transformed? You might want to pause the video here to have a go, and then we're gonna complete the sentences together.

So let's look at these triangles.

What can we say about the heights of the triangles? Well, let's look at the height of P, which is three units on our grid.

And the height of Q is six units on our grid.

So we can say that the height of Q is two times the height of P.

What about the width of the triangles? Well, the width of P is four units on our grid, and the width of Q is eight units on our grid.

So we can say that the width of Q is two times the width of P.

So we can say that we have a scale factor of two, linking triangle P to triangle Q.

Now you might be thinking there's a third side to those triangles.

There is, and it's much more more difficult to measure because it's across those diagonals.

But if we have scaled the height and the width by the same scale factor, two in this case, then we will have scaled the third side by the same factor, so those will be similar triangles.

So we can say that the dimensions of Q are two times the dimensions of P or the other way round, we can say that the dimensions of P are 1/2 times the size of the dimensions of Q.

Time for you to do some practise, we've got some more triangles here.

So you're asked to find out what scale factor has been used to change the dimensions of triangle A into the dimensions of triangle B? And they are similar triangles.

So A asks you to identify the scale factor.

B, is why you know this is the scale factor.

And then part C says draw a new triangle C, which is related to A with a scale factor of three.

And D says are triangles A, B and C similar? And can you explain your answer? So you might want to pause the video here so you can have a go and then we'll look at the answers together.

So let's have a look at these triangles.

So what scale factor has been used to change the dimensions of triangle A into the dimensions of similar triangle B? So let's have a look.

Well, we can see that if we look at the width of A, it's one square on our grid, the width of B is four squares on our grid, so the width has been multiplied by four.

So if the height of triangle A is three centimetres as it is, what must the height of triangle be B if they've been scaled by a factor of four? So it would be four times bigger, so the the height of B would be 12 units on our grid.

So if we look, we can see that indeed triangle B is 12 units high, so it has been scaled by a factor of four as well.

So the answer to B, I know this because the width and the height of triangle B are four times the size of the width and the height of triangle A.

So we know that we can link these two triangles with a scale factor of four.

C asked you to draw a new triangle C, which was related to A with a scale factor of three.

So what does that mean about the dimensions for our new triangle C? So these are related by scale factor of three to triangle A.

So if A was one unit in width, then if triangle C is related by a scale factor of three, triangle C must be three times as wide.

So it must be three units wide on our grid.

And we can see that triangle C that's drawn there is.

Triangle A is three squares or three units tall on our grid, so it has a height of three.

The new triangle must therefore be three times the height of A, so nine.

And if we look, we can see that triangle C is indeed nine centimetres if we've drawn it on centimetre paper, nine squares tall.

So your triangle for C should be three centimetres wide and nine centimetres tall, you may have drawn it in a different orientation, twisted it round somehow, drawn it the other way round.

I've drawn mine so it sort of matches.

And we can see that it's very much related to triangle A.

Part D said are triangles A, B, and C similar and explain your answer.

And we can say that, yes, they are because we can transform each one into the other by the use of a scale factor.

And we'll learn more about this as we go into the next part of our lesson.

Okay, so in the first part of our lesson, we've been identifying the scale factor in irregular polygons.

What we're now going to do is think about how we can express that scale factor as a ratio using the language of ratio.

So let's have a look at these shapes and think about the ratios between their sides.

Okay, so we're back with our rectangles A and B, and we know that there's a scale factor of three because the dimensions of A have all been multiplied by three to create triangle B.

So the dimensions of B are all three times the size of the dimensions of A.

We can think about that with that ratio language of for every.

So for every one centimetre in the sides of A, there are three centimetres in the sides of B.

Let's just have a think about that.

For the height of A, that's quite straightforward, isn't it? Because the height of A is one centimetre, and so for that one centimetre in B, there are three centimetres.

So for every one centimetre in the height of A, there are three centimetres in the height of B.

How does that work then for the width? If the width of A is two centimetres, well that for every one centimetres there are three centimetres.

If we double that then, if I've got twice as many centimetres in the width, I need twice as many centimetres in the width of B.

So I've doubled my one centimetre to two centimetres.

I've doubled my three centimetres to six centimetres, and we can see that the width of B is six centimetres.

So we can say that the ratio between the length of the sides in rectangles A and B is for every one centimetre in the sides of A, there are three centimetres in the sides of B.

We can make that into a sort of slightly shorter sentence and we can say that the ratio of sides A to B is 1:3 for every one centimetre in A, there are three centimetres in B.

And sometimes we write this with the to replaced by a colon.

So A to B is 1:3, but we must remember that what we are really saying there is for every one centimetre in A, there are three centimetres in the sides of B.

And we also looked at that transformation going the other way.

So if we started with B and transformed it into A, we had a scale factor of 1/3.

So for every three centimetres in the sides of B, there is only one centimetre in the sides of A.

So we are dividing by three each time.

So this time we can say that the ratio of sides B to A is 3:1, for every three centimetres in B, there is one centimetre in A.

And again, we can replace those words to with a colon, so we can say B two A is 3:1.

So why don't you have a go at using that language of ratio to think about the relationship between triangles P and Q.

So you might want to pause the video here and have a go at completing these sentences and then we'll have a look at them together.

Okay, so A asked us to identify the scale factor and we know that the scale factor between triangles P and Q is two.

If we multiply the dimensions of triangle P by two, we get the dimensions of triangle Q, so the scale factor is two.

How can we think of that as a ratio? Well, we might have started by saying that for every three centimetres there are in P, there are six centimetres in Q.

And that would be okay, but we can make that even simpler.

So if we think about that ratio of for every three in P, there are six in Q.

If we think about reducing the P value to one, so saying, well, what could we say for one centimetre? We've got that idea, remember of dividing of making it three times smaller.

So if I divide my three centimetres in P by three I get one centimetre, if I divide my six centimetres in Q by three, I get two centimetres.

So I can simplify that and say that for every one centimetre in the sides of P, there are two centimetres in the sides of Q.

And if you have a look at that, imagine that top centimetre of the height of P becomes two centimetres in the height of Q, the middle one becomes two, the bottom one becomes two.

For every one centimetre in the sides of P, there are two centimetres in the sides of Q.

So we can then say that the ratio of sides P to Q is 1:2, or we can use that colon long notation as well.

We could also do it the other way round and say that for every two centimetres in the sides of Q, there's one centimetre in the sides of P.

So the ratio of sides Q to P is 2:1, and it's actually quite important to make sure we get those relationships the right way round.

So it's useful to read questions really carefully to make sure that we know which way round we are thinking about that ratio.

Okay, time for you to do some practise.

So we've got a task here involving our triangles.

So we are gonna think about the scale factor and then we're gonna think about that language of ratio.

So you might want to pause the video here so you can have a go at this before we look at the answers together.

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

A asked us what the scale factor was between triangles A and B.

So let's have a look at that width of A, the width of A is one unit on our grid and the width of B is four units on our grid, four times bigger.

So we might be thinking that we've got a scale factor of four, but if it's a true scale factor of four, then the height of A must be multiplied by four to give us the height of B.

So the height of B must be four times the height of A.

So the height of A is three units multiplied by four would be 12.

And yes, as we check the height of B is 12 units.

So we can say that the dimensions of A have been multiplied by four to create the dimensions of B.

So the scale factor is four.

Part B asked us to think about that language of ratio, didn't it? So we can say that taking that one centimetre width is quite handy for us, so we can do a one centimetre comparison.

For every one centimetre in the sides of A, there are four centimetres in the sides of B.

And we can say that that is true for all the dimensions.

So for every one centimetre in the height of A, there are four centimetres in the height of B.

So for C we can express that as a ratio, the ratio of sides A to B is 1:4, or A to B is 1:4.

What about thinking the other way around from B to A? So for every four centimetres in the sides of B, there are one centimetre in the sides of A.

So the ratio of sides B to A is 4:1, and then we can write it with that colon notation, 4:1.

Okay, so we've identified scale factors, we've expressed scale factors using ratio, and in this final part of our lesson we are going to use ratio tables to help us to solve problems involving scale.

So let's have a look at some problems. Okay, so we're looking at our rectangles A and B again, and we've got more ways now that we can describe the relationship between shapes A and B, we can use scale factors and we can use ratios.

So we know that the scale factor from A to B is three.

The dimensions of A have been multiplied by three to give us the dimensions of B.

Or we can say that dimensions of B are three times the size of the dimensions of A.

We can also say that the ratio of sides A to B is 1:3.

For every one unit in the sides of A, we've got three units in the sides of B and that works for the height and the width, that's why they are similar shapes related by a scale factor.

If we think back the other way, we can say that the dimensions of B have been divided by three to create the dimensions of A, we can say that A is 1/3 the size of B, the dimensions of A are 1/3 the size of the dimensions of B.

So the scale factor from B to A is 1/3.

And as a ratio we can say that the ratio of sides B to A is 3:1, or we can use that colon note notation again.

So how could we represent this information in a ratio table? And you might have come across ratio tables to help you to organise information in order to solve problems involving ratio and scaling in terms of maps and plans and things like that.

So it's about organising the information that we've got in order to see those relationships and at some point use that to solve some problems around missing dimensions in shapes.

So let's have a look.

So I've created a little table here for rectangle A and rectangle B and their height and their width.

If I fill in these values, it will help me to identify the relationships.

So let's have a look at rectangle A, that first column.

Its height is one unit and its width is two.

Rectangle B, its height is three units and its width is six units.

So you can see here that I've got A multiplied by three, one multiplied by three is equal to three.

So the height of A has been multiplied by three to equal the height of B.

I could work backwards and I could say that the dimensions of B have been multiplied by 1/3 or divided by three to create the dimensions of A.

So I can see those relationships, I can see those ratios in a way in my table.

Okay, so let's have a go at another one.

Here are our triangles A and B again.

So can you complete the ratio table to show the scale factor between triangles A and B? So you might want to pause the video here and have a go yourself and then we'll have a look at it together.

So let's put those values in.

The height of triangle A is three units and the height of triangle B is 12 units.

The width of triangle A is one unit and the width of triangle B is four units.

You might have filled it in for one triangle at a time.

At that time I went across and did the heights of both and the widths of both, so then we can spot those relationships, can't we? So we can see that there's a multiplying by four to transform triangle A into triangle B, three multiplied by four is 12, one multiplied by four is equal to four.

But then we've also thought about scaling down, going back the other way.

So we can see that we've got that 1/4 link, division by four if we transform B into A.

And we've been talking about scale factors, we need to know where that scale factor is.

That scale factor between triangles A and B is four.

It's the number that we need to multiply the dimensions of A by in order to get the dimensions of B.

So that's where our scale factor is in our ratio table.

Within the table we've got the dimensions of the shapes and that relationship between the dimensions, that multiplication that we have to do, gives us our scale factor.

Equally, we said that when things get smaller, that scale factor is a fraction.

So we've got the scale factor of 1/4 transforming B back into shape A.

There's another relationship we can look at when we look at these triangles.

We can look at the relationship between the height and the width within a triangle, and this can be really useful when we're solving problems. So let's have a think.

What do you notice about the width of triangle A and the height of triangle A, what would the relationship be there and what would the relationship be between the width and the height of triangle B? What do you notice? Well, I hope you've noticed that they're both multiplied by three and that's really crucial because we know that those triangles are similar triangles, that they are related by a scale factor, not only because their dimensions have been multiplied by the same factor in order to transform one into the other, but that the relationship between their height and width has stayed constant.

The relationship between the height and width of triangle A is multiplied by three, and the relationship between the height and the width of triangle B is represented by multiplying by three.

So that relationship has stayed the same.

And that's a really useful one to think about.

You might need it when we come to some tasks at the end of this lesson, and as we've just said, they're similar triangles.

So the relationship is the same and you can use this ratio or scale relationship to solve problems. Okay, time for some practise, we've got three rectangles here, A, B, and C, and we've got some missing dimensions to calculate.

We've got the height of rectangle A and the width of rectangle C to calculate, we are told that they are similar rectangles.

So this means that they're all related by scale factors and as we've just learned that we can also say that the relationship between their own height and width will be constant across all three triangles.

So your task is to calculate the missing dimensions and to use ratio grids to record your working.

So you might wanna pause the video here and then we'll look through together.

So you can see here that I drew a ratio table.

So I had rectangle A on the left and B on the right.

And then I've got height of the rectangle is my first row and width of the rectangle is my second row.

So the information I've got is, I know that the height of rectangle B is 24 centimetres and the width is eight centimetres.

I don't know the height of rectangle A, but I know that the width is four centimetres, but I know that they are similar.

So they are related by a scale factor.

So can I work out what that scale factor is? Well, I could work out the relationship between the height and the width within the rectangle.

So I've got here that the relationship between the width of B and the height of B is whatever connects eight with 24.

And remember I'm thinking multiplication because I'm thinking about those scale factors.

Well, I know that eight multiplied by three is equal to 24, so the width of triangle A multiplied by three gives me the height.

The height of triangle B is three times its width.

So if these rectangles are similar, then the height of triangle A must be three times its width.

So if I multiply four by three, I will get 12.

But is there another way we could work this out? Let's have a look and see if we get the same answer thinking about it in a different way.

I also know that there's a scale factor that has transformed rectangle A into rectangle B.

And if I look at the widths of the rectangle, I might be able to work that out.

So the width of the rectangle A is four centimetres and B is eight centimetres.

So what is my factor that links those? Well that's multiplied by two, isn't it? So the width of rectangle B is two times the width of rectangle A.

And if they're similar then I know that same relationship will apply to the height.

The height of rectangle B must be two times the height of rectangle A.

But if it's two times, then I need to go the other way so I could divide by two.

And if I divide 24 by two, I get the answer 12.

I could also think of it as multiplying by a half.

But whichever relationship I use, I know that the height of rectangle A is 12 centimetres.

Which relationships did you spot? Which ratio did you use to calculate your answer? Did you use the scale factor between A and B or did you use the relationship within the rectangles? So the the height to width ratio within both rectangles, which we know is constant because they're similar.

And part B of this wanted us to find the width of rectangle C.

So again, I've drawn a ratio table, I've gone for rectangle A to help me here.

I wonder if you went for rectangle A or rectangle B, but I thought rectangle A might help me here.

So I've put in the information for rectangle A.

I now know that the height of rectangle A is 12 centimetres, I've just calculated that, and the width of it I knew was four centimetres.

I know that the height of C is 600 centimetres, it's a lot bigger, it did say these weren't drawn to scale.

So rectangle C would be an awful lot bigger if it was drawn on square paper or on centimetre squared paper.

But I know that its height is 600 centimetres, but I don't know what its width is.

So what relationship should I use in this table to help me work it out? Well, let's have a look at the different things I could think about.

If I think about the relationship within the rectangles.

If I think about the relationship between the width and the height of the rectangle, I can see that in rectangle A the width is four centimetres and the height is 12 centimetres.

So the height of the rectangle is three times the width of the rectangle.

So that must be the same for rectangle C.

So if I divide my height by three, I will work out the width of my rectangle and 600 divided by three is equal to 200.

But I could also think about it a different way, couldn't I? I could think about the scale factor between rectangles A and C.

Wow, I've got to think about what would I multiply 12 by to equal 600.

That's quite a lot to think about, isn't it? There's a 60 and 12 I quite like there because five times 12 is 60, so this must be 50 times right? So the height of rectangle C, 50 times the height of rectangle A.

But that does mean then that the width of rectangle C must be 50 times the width of rectangle A and four multiplied by 50 is equal to 200.

The same dimension that I got obviously if I divided 600 by three.

So again, I wonder which relationships you spotted, which ratio did you use to calculate your answer? And I think you can see from that example that sometimes it's more efficient to look at the relationship within a pair of similar shapes than it is just to look at the scale relationship and look at the scale factor that transforms one into the other.

And we've come to the end of our lesson.

Thank you for all your hard work.

We've looked at scale factors again and worked out that we can describe scale factors whilst also thinking about ratio that the two are very much linked and that ratio tables that we might have used to solve problems in other contexts can be used to solve problems involving scaling.

You can now identify scale factors by finding relationships between shapes and relationships between the sides within shapes.

And we've looked at that now with irregular polygons.

We know also that similar shapes are connected by scale factors and have the same relationship between the sides within the shape.

So similar shapes have a scale factor that transforms one into the other, but also have the idea of a scale factor within their own dimensions, which will stay constant as we create similar shapes.

Thank you for your work today and I hope to see you again.

Bye.