Lesson video

Today we'll be learning to solve problems involving scale factor in shape.

All you'll need is a pencil and piece of paper in today's lesson.

So grab your things if you haven't done so already.

Here's our agenda for today's learning.

We're going to solve problems involving scale factor in shape.

We don't have a knowledge quiz today.

We're going straight into investigating congruent in similar shapes.

Then we will look at describing the relationship between similar shapes, before you do some independent learning, and then a final quiz.

So we're going straight into having a look at some shapes.

I would like you to have a think about what you notice about these shapes.

What is the same about them and what is different? Pause the video now to make some notes.

So what you will have noticed is that the mathematical properties of all of these shapes are exactly the same.

These shapes are congruent.

They have the same length sides and they have the same internal angles.

So the word for these shapes is congruent.

The only thing that is different about them is that orientation.

Some of them have been twisted around in different ways.

Now have a look at these shapes and think about what you notice and make some notes.

So these shapes are not congruent, but they do share specific similarities.

So they do have the same size internal angles, and they have the same relationship between the respective side lengths.

But the side lengths have changed.

So the shapes have become bigger or smaller.

So we wouldn't say that these shapes are congruent, but we would say that these shapes are similar.

But what do we mean by similar? So I've got two shapes here which are similar and two shapes which are not similar.

So have a look at the first set of shapes.

These are similar because corresponding angles are equal.

So you can see this angle up at the top left hand corner is 116 degrees as is this one in the smaller shape.

All of the corresponding angles are the same.

The only thing that is different is that the side lengths are different, but they have been enlarged by the same scale factor.

So you can see that this shape at the bottom has been enlarged to make the shape at the top.

And the side length is three centimetres.

The top side length is six centimetres, which means that the side length has been doubled.

And the same can be also said for the bottom, which was four.

And that's been doubled to be eight centimetres.

So they've been enlarged by a scale factor two.

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

That means that the relationship between the pairs of sides is the same in these two shapes.

That's why these are similar shapes.

These shapes are not similar.

The corresponding angles are completely different, and they have not been enlarged by the same scale factor.

So these ones, we would not even describe as similar.

Even though they are both trapezial, there's nothing else similar about them.

So have a look at these two triangles.

I want you to think about whether they are similar or not.

So our top one has a base of three centimetres and a height of 12 centimetres.

The bottom one has a base of two centimetres and a height of six centimetres.

Youcef thinks the triangles are similar because they're both triangles and because six here, the height is half of 12.

But Elizabeth says, six is half of 12, but two is not half of three.

So she thinks they're not actually similar.

What do you think? Pause the video and make some notes.

So while Youcef is correct, they are both triangles, and six is half of 12, the relationship between the bases would have to be the same.

So we would have to have a base which was half of the other base.

So this, we would be looking at this one being double this one.

So it would have to be 1.

5 centimetres.

So therefore they're not similar because they haven't been enlarged by the same scale factor.

So Elizabeth is correct in this case.

Now we're going to look at some more similar shapes, and we're going to describe this relationship in more detail.

So Ronaldo drew two similar triangles.

How can we describe the relationship between triangle A and triangle B? So we can see that these are both right angle triangles.

Triangle A has a base of four centimetres, and B has a base of eight centimetres.

and A has a height of six centimetres and B has a height of 12 centimetres.

So you can see that the base of the triangle B is double the width of triangle A, and the height of triangle B is double the height of triangle A.

So we can say that Ronaldo has enlarged triangle A by a scale factor of two.

He's multiplied both lengths in both sides by two.

Therefore, he's enlarged it by scale factor two.

Now I want you to think about how we can write this relationship as a ratio.

So think about the length of the base, we can use that one to help us and the length of the base in B, how would you write that as a ratio? So we could write the ratio as four to eight, which simplifies as one to two.

And you can see the relationship between these is that the number in the second part of the ratio is double that of the the first.

So it's been multiplied by two or increased by scale factor two.

Let's look at another one together.

What would be the base and height measurements of triangle B if triangle A were enlarged by a scale factor of six? So if it's enlarged by a scale factor six, that means that we multiply the base by six, and we multiply the height by six.

So the base in B would be four multiplied by six, which is equal to 24 centimetres.

And the height of B would be six multiplied by six, enlarged by a scale factor of six, which is 36 centimetres.

And then we're going to write the scale factor as a ratio.

So I'm going to use the base lengths to help me.

So I can write that as four to 24, which simplifies as a ratio of one to six.

So if we're enlarging by a scale factor of six, we're multiplying by six.

And you can see the relationship between the two numbers within a ratio.

The second number is six times greater than the first number.

Now it's your turn.

So I want you to think about in this triangle, what would the base and height measurements be if triangle A was enlarged by a scale factor of 10? And how could we write the scale factor as a ratio? Pause the video and answer the questions.

So increased by a scale factor of 10, the six centimetre height would become 60 centimetres and the four centimetre base would become 40 centimetres.

So writing that as a ratio, we would look at the base and say that's four to 40, which is a ratio of one to 10.

And if you look at the relationship within the ratio, four times 10 is 40, one times 10 is 10.

Now it's time for you to complete some independent learning.

So pause the video, complete the tasks, and then click restart once you're finished.

So for question one, you were given a triangle not drawn to scale, and asked what its dimensions would be if the triangle was enlarged by scale factor five.

So that means multiplying each dimension by five.

So I'll start with the three centimetres, three centimetres multiplied by five is 15 centimetres, two centimetres multiplied by five is 10 centimetres, and four centimetres multiplied by five is 20 centimetres.

Question two, rectangle B is an enlargement of rectangle A.

What is the scale factor of the enlargement? So you were looking for the relationship between the sides.

So the vertical side was two centimetres on A and four centimetres on B.

So that's doubled, it's multiplied by two.

Check for the base.

Four centimetres to eight is also doubled.

Multiplied by two means that it has been enlarged by scale factor two.

For question three, you were asked to find the dimension, the missing dimension on the enlarged triangle and write the scale factor of the enlargement.

So you needed to use the two corresponding sides where you knew the measurement.

So five multiplied by four is equal to 20.

So you know that it's an enlargement of scale factor four.

So you're multiplying 2.

5 by four, which gives you 10 centimetres.

For question four, Kyra has drawn a picture of rhinoceros, which is four centimetres tall and seven centimetres long.

She wants to put her design onto the cup and the billboard, so that it has just enough space inside, maybe a little bit leftover.

So by what scale factor will she have to enlarge the picture by to fit on the cup? We can see that the relationship between four centimetres and 12 centimetres is that four multiplied by three equals 12, seven multiplied by three is equal to 21.

So it will just fit in that with one centimetre leftover.

So that would need to be enlarged by scale factor three.

For the billboard, we've got different units here.

So four centimetres and eight metres, I'm going to convert the metres into centimetres.

So that will be 800 centimetres.

Four times 200 is equal to 800 centimetres.

And this would be 1,400 centimetres.

Seven times 200 is equal to 1,400 centimetres.

So that would fit very snugly if it was enlarged by a scale factor 200.

Great work today.

Well done.

Don't forget to complete your final quiz before you finish today's lesson.