Lesson video

Hello, my name is Dr.

Rolandson, and I'll be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the Unit of Transformations.

This lesson is called Introduction to Enlargements, and by end of today's lesson, we'll understand the nature of enlargements and appreciate what changes and what is invariant when an enlargement occurs.

This lesson contains two keywords.

The first is enlargement, which is a transformation that causes a change in size.

The second is similar, and two shapes are similar if the only difference between them is their size.

Also, the side lengths are in the same proportions.

The lesson will have two learn cycles, with each learn cycle exploring one of those two key words.

And in the first learn cycle, we'll go understand enlargement.

Enlargement is a transformation that causes a change in size, and that could be any change in size.

It could be a stretch, or it could be a shrink.

They're both considered as enlargements when we talk about transformations.

For example, we could start with an object, apply a transformation to it that causes the image to be bigger in size, and that is considered to be an enlargement.

Or we could start with an object and apply a transformation that causes the image to be smaller in size, and that is also considered to be an enlargement.

One thing to know is that enlarging a shape does not affect the size of the angles.

So for example, here we have a right angle triangle with other angles, 30 degrees and 60 degrees.

If you imagine enlarging each of the lengths of that triangle, imagine the horizontal line getting longer and the vertical line getting longer as well.

The angle between those two sides will still be 90 degrees, and if we enlarge all those lengths by the same amount, then all the angles will still be the same.

We'd have a 90-degree angle, a 30-degree angle, and a 60-degree angle just like we did in the object.

The angles, therefore, are invariant during an enlargement.

So, let's check what we've learned so far.

What type of transformation has occurred in this situation here, from the object to the image? Is it A, an enlargement; B, a reduction; C, a rotation; or D, a reflection? Pause the video, have a go, and press play when you're ready.

The answer is A.

An enlargement has occurred.

The image is a different size to the object.

What type of transformation has occurred this time, from the object to the image? Is it A, an enlargement; B, a reduction; C, a rotation; or D, a reflection? Pause the video, have a go, and press play when you're ready for the answer.

The answer is A, an enlargement again.

It is a change in size.

It does not matter that it's got smaller.

Any change in size is considered to be an enlargement when we talk about transformations.

Which property is always invariant during an enlargement? Is it A, the lengths are always invariant; B, the angles are always invariant; or C, the areas are always invariant? Pause the video, have a go, and press play when you're ready to continue.

The answer is B.

The angles are invariant during an enlargement.

The angles do not change when the shape gets bigger.

And finally, for our check for understanding, the object has been enlarged to create the image.

What is the size of the angle marked with a question mark? Pause the video, have a go, and press play when you're ready for the answer.

The answer is 44 degrees.

The angle is the same as the corresponding angle in the object.

The angle hasn't changed, so it's still 44 degrees in the image.

How did we get on? Okay, it's over to you now for task A.

This task contains two questions.

Question one, you need to write true or false for each of the statements below.

Pause the video, read each statement, write true or false, and press play when you're ready for question two.

Here's question two.

Alex is going to enlarge the parallelogram to double the size.

He thinks that the corresponding angle in the image will be 70 degrees.

Do you think he is correct? Justify your answer.

So, in this question, write yes or no about whether or not you think he is correct.

And for either answer, write a sentence that explains why you think he's correct or why you think he is incorrect.

Pause the video, have a go, and press play when you're ready for both of the answers.

Well done, everyone.

Let's take a look at these statements for question one.

A, enlargement only refers to when a shape increases in size.

That is false because it could be any change in size, whether it's a increase in size or a decrease in size.

Statement B, the image may have different lengths to the object after an enlargement.

That statement is true.

Statement C, the image may have different angles to the object after an enlargement.

That statement is false.

Remember, the angles are invariant during enlargement, which means they don't change.

And statement D, the image can be smaller than the object after an enlargement.

That statement is true because an enlargement can be any change in size, whether bigger or smaller.

And question two, do you think he's correct? Justify your answer.

No, Alex is not correct, and here are some examples of reasons you could give to justify your answer.

You could say that enlarging the shape does not cause the angles to change size, or you could explain that in other words by saying the angles are invariant during an enlargement.

Alternatively, you could say what the angle would be in the enlarged image.

You could say that the corresponding angle will be 35 degrees.

Anything along those lines is absolutely great.

Well done, everyone.

You're doing great.

Onto the second cycle for today's lesson, which is about understanding similarity.

Here's a photograph of Lucas, and here are three larger versions of that same photograph, but there's a problem with the first two photographs because they look a bit distorted.

In the third photograph, Lucas's face looks the same as it does in the original one but not in the first two.

Can we think why does Lucas's face look distorted in the first two images but not the third? The reason is because the third image is similar to the original photograph, but the other two aren't.

Let's explore why that's the case.

Two shapes are similar if they are in the same proportions.

Now, let's explore what that means by looking at an example.

Here we have two versions of the same photograph, a small version and a larger version, and we can see that they look pretty much the same apart from one's bigger than the other.

In the first one, in the smaller image, we can see that the width, six, is three times the height, which is two.

Let's take a look at the bigger image.

We can see the width of 12 is also three times the height, which is four.

Because those multiplies are the same, because the width is three times the height in both of those images, they are in the same proportion.

The rectangles are similar because the multipliers within each shape are equal to each other.

They're both three.

We can look at this in a different way.

If we compare the small one to the big one, side for side, so for example, if we look at the height for each photograph, we can see that the height of the large photograph is two times the height of the small photograph.

Two times two is four.

And we look at the widths of each photograph, we can see that the width of the large photograph is also two times the width of the smaller photograph.

So the multiplier from the small to the large is two for both the height and the width.

Therefore, they are in proportion.

So, the rectangles are similar because the multipliers between each shape are equal.

Two different ways to explain the same idea.

Let's clarify that now with a non-example.

We look at these two photographs, we can see that they are not similar because the photograph on the right, the larger photograph, the faces look a bit taller and a bit thinner than they do in the smaller photograph.

Let's see why that is the case.

If we look at the measurements within each photograph, so for example, to get from the height to the width in the smaller one, we multiply by three, but in the larger photograph, can see that to get from the height to the width, it's not multiplied by three this time.

It's multiplied by two.

The width is double the height in the large photograph, but it's three times the height in the smaller photograph.

Because those multipliers are different, it means they are not in the same proportions.

Therefore, the rectangles are not similar because the multipliers within each shape are not equal to each other.

Let's look at this non-example in the other way.

We can see we compare the heights for each one.

To get from the smaller photograph to the larger photograph, we times by three.

But we look at the widths of each photograph, we can see that to get from the smaller photograph to the larger photograph, we times by two.

Those multipliers, again, are different.

Therefore, they are not in the same proportions.

The rectangles are not similar because the multipliers between each shape are not equal to each other.

Let's explore this idea of similarity a bit more now by doing a question together.

Which triangle is similar to triangle A? So we've got three triangles.

A is on the left, and either B or C is similar to triangle A.

Let's have a think about how we could go about finding the answer.

Pause, have a think about it, and then press play when you're ready to continue.

Okay, let's do this together.

There are a couple of different ways we could do this.

We could look at the multipliers within each triangle, but the problem here is, if we look at triangle A, the multiplier from three to four, it's gonna be a fraction or a decimal.

Now that's fine.

We can work with fractions and decimals.

We could use a calculator if we want to.

But there might be an easier way.

If, instead, though we look at the multipliers from one shape to another, that might be a bit easier.

In this case, four multiplied by three makes 12, and three multiplied by three makes nine.

We can see here that the multipliers are the same for both the height and the width of the small triangle to get to the height and width of the bigger triangle in B.

Therefore, B must be similar to A.

Let's double check this by having taken a look at C.

Four multiplied by three is 12, and three multiplied by two is six.

Therefore, C is not in the same proportions as A.

Also, C is not similar to A.

Our answer to this question is B.

Let's do another one.

Which triangle is similar to triangle A this time? Again, have a little think about how we might approach this question.

Pause, have a think, and press play when you're ready to continue.

Okay, let's do this together.

Now, we could, again, look at the multipliers to get us from A to B, but the problem at this time is the same as what we had last time.

Three times something makes four.

Well, that's gonna be a decimal, or it's gonna be a fraction, which makes life a bit trickier than it might need to be.

So, let's look at the multipliers within these triangles and see if it's a bit easier that way.

Three times three makes nine.

So, let's take a look at B and C and find out if it's the same multiplier in those as well.

In triangle B, four times three makes 12, so that's good; that works.

But in C, hmm, six times two makes 12.

So, that triangle is not in the same proportions as A because that multiplier is different, but B is similar 'cause they have both a multiplier of three.

So, answer is B again.

Okay, let's check how well we've understood what we've just done.

Which rectangle is not similar to the object? We have the object at the top of the screen and three rectangles, A, B, and C, and I want to know which of those three rectangles is not similar to the objects.

Pause the video, have a go, and press play when you're ready for the answer.

Okay, we can get this by looking at each rectangle one at a time.

With rectangle A, we can see that the width, six, is double the height, three, and in the object, the width, four, is also double the height, two.

Therefore, in both A and the object, the width is double the height, so they are in the same proportions, so they are similar, which means A is not our answer.

Rectangle B, we can see, again, the width, two, is double the height, which is one.

Therefore, B is similar to the object and, therefore, is not our answer.

In C, we can see the rectangle has a width, three, which is three times the height, which is one, whereas in the object, the width is two times the height.

Therefore, rectangle C is not similar, and that is our answer.

Earlier, we saw that enlarging a shape does not affect the size of the angles.

Now, when we consider that two shapes are similar if one is an enlargement of the other, it means that corresponding angles in similar shapes are also equal.

Let's explore that through a example and a non-example.

Here we have an example of two similar triangles, and we can see that they're similar because the lengths in the smaller triangle, three and four, have both been multiplied by two to make the lengths in the larger triangle, six and eight.

Now, let's look at the angles in each triangle.

We have a 90-degree angle, a 53-degree angle, and a 37-degree angle in each triangle.

Therefore, the angles in those two similar triangles are the same.

Corresponding angles are equal in similar shapes.

This non-example is two triangles that are not similar, and we can see that they're not similar because I've multiplied the three by two to get the six, whereas I've multiplied the four by three to get the 12.

I've not multiplied by the same thing.

So, now let's look at the angles in these two triangles.

Okay, I've got 90 degrees.

They're the same, but hmm, 53 degrees and 63 degrees, that's different.

And 37 degrees and 27 degrees, they're different as well.

Therefore, the corresponding angles are not equal in these two triangles, which are not similar.

Let's check we've understood that.

Which triangle is similar to the object? Triangle A, B, or C? Pause the video, have a go, and press play when you're ready for the answer.

The answer is B.

In triangle B, all three of those angles are the same as in the object: 54, 73, and 53.

Let's check that in a different way.

Which triangle has the same angles as in the object? Is it triangle A, B, or C? Pause the video, have a go, and press play when you're ready for the answer.

The answer is A.

The way we can get this is by looking for which triangle has its lengths in the same proportions because that would mean that triangle is similar to the object.

We can see that the lengths of three and four in A are both half the lengths of the object, which are six and eight, whereas in triangle B, the length, two, is a-third of the length, which is six in the object, whereas four in triangle B is only half the length of the object, which is eight.

Hmm, so they're not similar.

And to get from the object to triangle C, well, the three is half of the six, but the six is not half of the eight.

It doesn't matter what that is.

What does matter is six is not half of eight.

So 'cause it's not half in both occasions, it means that C is not similar to the object.

Therefore, the only one with the same angles is triangle A.

Okay, over to you for task B.

This task contains two questions.

In question one, you've got three rectangles on the left and three rectangles on the right, and we need to match up which rectangles on the left are similar with which rectangles on the right.

You can do that by drawing lines from one rectangle to the other.

Pause the video, have a go, and press play when you're ready for question two.

And question two.

In this question, you'll need some square paper.

Use a square paper to accurately draw the right angle triangle below.

So you want a one length, which is four, and another length which is three, of a right angle between them and use the squares to help you with that right angle, and then, just join up that other length, which goes diagonal there.

And then, draw another right angle triangle on your square paper, which is similar to this one.

Then, cut out each triangle, and then, place one triangle on top of the other, preferably the smaller triangle on top of the big one, and match up one of the vertices.

So place the triangles, the corners, the vertices, are both overlaying each other, and see what you notice about the triangle and those vertices.

And then maybe try different vertex, maybe another vertex, and see what you notice each time when you move the smaller triangle around the bigger triangle.

Pause the video, have a go, and press play when you're ready for the answers.

Okay, let's see how we got on.

Question one.

It said match the rectangles on the left with the ones on the right.

So, the top rectangle on the left matches with the second rectangle on the right.

We can see that even by thinking one times three makes three in the small rectangle, and four times three makes 12 in the large rectangle.

It's the same multiplier each time, times three.

Therefore, they must be similar.

Or we can see it by multiplying this lengths in the small rectangle by four each time to get lengths in the big rectangle.

One times four is four.

Three times four is 12.

It's times four each time.

Either way, those two rectangles are similar.

The second rectangle on the left matches with the bottom rectangle on the right.

The easiest way this time is probably to look at what you multiply the left rectangle by to get the right rectangle.

We can see that four times a-half makes two, and 10 times a-half makes five.

'Cause it's times a-half both times, it means that the rectangle on the right is similar to the one the left.

And the bottom rectangle must match up with the top one.

Here, we can see the probably easiest way to do this is to look at what I multiply eight by to get 16 in the left rectangle, and multiply eight by two, and see if it's the same as in the right-hand rectangle.

Six times two also makes 12.

It's times two in both cases, so they must be similar.

Question two.

In this question you need to start by drawing that triangle on some square paper and then draw another right angle triangle, which is similar to that one.

The way we can check our answer's correct is by looking at wherever the base and height, there should be a pair of numbers that is obtained by multiplying four and three by the same number.

So for example, if you've got a base of eight and a height of six, four and three have both been multiplied by two, so that's great.

Or you can have a base of 12 and a height of nine, or a base of 16 and a height of 12, and so on.

Or it could even be smaller, a base of two and a height of 1.

5.

So long as both four and the three have been multiplied by the same thing to get your base and height, it is similar.

And then, you have to cut out each triangle and place one on top of the other to take a look at what happens when you match up the vertices.

You may have placed one on top of the other in any of these three different ways, and hopefully, what you may have seen in that process is that the vertices completely cover each other perfectly, which means the angles at those vertices are perfectly the same.

So in that first example, it's the angles in the bottom left-hand corner which you can see match up perfectly 'cause they are the same.

In the middle one, it's a top corner which match up perfectly.

Those angles are the same.

And the right-hand example, those two right angles in the bottom right-hand corner, they match up perfectly, so you can see that those two angles are the same as well.

Hopefully, our conclusion is that corresponding angles are equal in similar shapes.

Well done, everyone.

Here's a summary of what we've learned in today's lesson.

Enlargement is a transformation that causes a change in size, and that can be any change in size.

It could be a stretch, or it could be a shrink.

They're both considered enlargements when we talk about transformations of shapes.

Angles remain invariant when a shape is enlarged.

The lengths and areas may change, but the angles always remain the same in the image as in the object when an enlargement has taken place.

If two shapes are similar, it means that the corresponding angles are equal and the lengths are in proportion.

In other words, the multipliers within the shapes or the multipliers from one shape to the other are the same throughout.

And when a shape is enlarged, the image is similar to the object.

Great job today, everyone.

Well done.