Lesson video

Hello, my name is Dr.

Ralston and I'm delighted that you'll be joining me in today's lesson.

Let's get started.

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

This lesson is called Enlarging Objects.

And by the end of today's lesson, we'll be able to enlarge objects using information about the centre of enlargement and scale factor.

Here's a reminder of some keywords that we have seen before and will be reusing in today's lesson.

This lesson includes two learn cycles.

In the first cycle, we'll be enlarging shapes using integer scale factors.

And the second, we'll be using non-integer scale factors.

Let's make a start at enlarging shapes using integer scale factors.

Let's learn how to enlarge a shape by an integer scale factor by looking at this example.

Enlarge a triangle by a scale factor of 2 from the centre of enlargement.

The great thing about enlarging shapes is that there are lots of different ways that you can do it.

So let's look at three different ways of solving this example together.

One method could be to measure with a ruler.

So we'll draw a line through a vertex from the centre of enlargement.

And using a ruler, measure the distance between the vertex and the centre, which we can see here is 5 centimetres.

Multiply that distance by the scale factor to get 10 centimetres.

And mark the corresponding vertex in the image, which we can see in the diagram.

We then repeat that for each other vertex.

So, draw a line from the centre of enlargement for the middle vertex, measure it, it's 3 centimetres.

Times it by the scale factor to get 6 centimetres, and mark that one.

Draw a line for the third vertex.

Measure it to get 5 centimetres.

Times it by 2 to get 10 centimetres, and mark it on.

And then finally, join up the new vertices.

A second method.

If we have a grid in the background, we can do this by counting squares.

Choose a vertex on the object.

Starting at centre of enlargement, travel along the grid to get to the vertex.

So you can see here, we've gone down the grid, then across.

Count the number of squares you travel in each direction.

We've travelled 3 squares down and 4 squares across.

Multiply that by the scale factor of 2, in this case.

And we get 6 squares down and 8 squares across.

Mark on the corresponding vertex.

And repeat that process for each of the other vertices.

So in the middle vertex, we travel just 3 across.

Times it by 2.

And we travel 6 across.

In the top vertex, we travel 3 up, and then 4 across, times it by 2.

6 up and 8 across.

Join up all the new vertices and we have our image.

A third method could be to find a position of one vertex, and then use that to find the positions of the others.

So we could use any method, whether that be using a ruler or count squares to find the position of the first vertex in the image.

You can see that's marked on our diagram there.

And then, let's look at the object, and we can see that the distance between two vertices in this case is 6.

Multiply that by the scale factor and we get a distance of 12.

So we can find the new vertex in our image is 12 above the first one we plotted.

And then repeat that for the other vertices.

We can see, to get the third vertex, it is 1 to the left and 3 up from the first one we plotted.

Multiply that by the scale factor, and we get 2 to the left and 6 up from the first one we plotted, and then we can join up our vertices.

So we've seen three different methods of how to enlarge a shape.

And the one we use may be a matter of choice depending on the situation we're in.

Measuring with a ruler can be particularly helpful when there is no grid in the background.

Whereas if there is a grid in the background, then it may be easier to do it by counting squares.

But, in whichever situation we're in, we have a choice between finding the position of each vertex individually in relation to the centre of enlargement, or finding the position of one vertex, and then finding out the other ones in relation to that.

The choice is yours.

Let's check what we've learned so far.

Sam is enlarging the object below by a scale factor of 3.

They measure the distance from the centre enlargement to the first vertex.

How far will the corresponding vertex be from the centre of enlargement? Pause the video, have a go, and press play when you're ready for the answer.

The answer is 12 centimetres, which we get from doing 4 times a scale factor of 3, which gets 12.

The shape is being enlarged by a scale factor of 4, which is the correct position for the corresponding vertex.

Is it point A, point B, point C, or point D? Pause video, have a go, and press play when you're ready for the answer.

The answer is C.

To get from the centre of enlargement to that vertex, which is marked on the object, we go 2 to the right and 1 down.

If we multiply that by 4, we'll be going 8 to the right and 4 down, which would take us to point C.

Let's look at some more tips that can help us when enlarging a shape.

Sometimes the centre of enlargement is somewhere on the object.

When the centre of enlargement is at one of the object's vertices, the distance between that and its corresponding vertex on the image will be 0.

That means that point will be invariant.

That vertex will be in the same position on the object and in the image, which therefore means the shapes will overlap, like you can see here.

This next tip is about checking that the position of the image is correct in relation to the object and the centre of enlargement.

We've previously drawn lines from the centre of enlargement through the vertices of the object while we were enlarging the shape.

Those lines can also be used to check that the image is in the correct position, because all the vertices on the image should be on a line.

For example, if we enlarged a small triangle to get the big triangle by drawing lines from the centre of enlargement through each vertex of the small triangle, we can then see that all the vertices of the bigger triangle lie on those lines.

So, it is in the correct position.

Whereas in this case, we can see that the vertices for the big triangle are all too low, which means that they are not on the lines, and we can see that it's not in the correct position.

And this one, we can see that two of the vertices seem to be in the right place.

They're on lines, but the top vertex is not on the line.

Therefore, that vertex must be wrong, and is one to double check.

Let's check what we've learned.

True or false, the enlargement has been drawn correctly.

First, decide whether it's true or false.

Pause, write true or false, and press play when ready.

The answer is false.

Let's look at some justifications.

Here are two possible ways to justify your answer.

Pause, choose one, and press play when you've decided.

The correct justification is A.

Some other vertices in the image are not on a line.

Okay, over to you for Task A.

This task includes four questions.

The first two questions are on the screen now.

In question one, you need to enlarge the object by a scale factor of 3 from the point which is marked on the grid.

And in question two, you need to enlarge the object by a scale factor of 4 from the point with the coordinate (2, 1).

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

And here's questions three and four.

In question three, enlarge your object by a scale factor of 2 from the point that is marked, and there's no grid in this question.

And in question four, there's also no grid, where we need to enlarge your object by a scale factor of 3 from the point which is marked on the vertex there.

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

Great work.

Let's see how we got on.

This is what your answers should look like in questions one and two.

For each one, you need to check that the image is the correct size and in the correct position.

So in question one, you've got a right angle triangle.

Check where it's 3 squares across and 3 squares down, and that'll be the correct size.

And then check it's in the right position.

Look at the bottom vertex and check if it's 6 squares to the left of the centre of enlargement.

And in question two, we should have a parallelogram, which is 4 squares across, and has a height of 4 squares, which vertices in the coordinates, (2, 5,), (6, 5), (6, 1), and (10,1).

If you've got all those things correct, those two questions are perfect.

Well done.

And here is what our answers should look like in questions three and four.

Accuracy is key with this.

So here are some tips to make sure that it is done accurately.

Using a ruler, measure distance from the centre enlargement to the top right hand corner of the object.

And measure the distance from that same point to the corresponding point on the image.

Those two distances should be the same.

Also, you can check by measuring the length and the width of your image and checking up those are the same as well because it is a square.

In question four, we can measure the distance from the centre of enlargement to the top of the object.

And then measure the rest of the distance from there to the top of the image.

And just double check that the second distance, which we've labelled F there, should be equal to double the first distance, which we put as E there.

So F should be equal to 2e.

And do the same along the bottom as well.

First, measure the distance from the centre of enlargement to the rightmost vertex on the object.

And then measure from there to the rightmost vertex on the image.

And that second distance should be double the first distance.

And that would mean that the overall distance from the centre of enlargement to the vertex on the image is 3 times the distance from the centre of enlargement to that same corresponding vertex on the object.

If you're unsure, ask someone else to check it for you.

Well done so far.

Now let's learn to enlarge a shape using a non-integer scale factor.

Let's learn how to do this by looking at this example here.

Enlarge a triangle by a scale factor of 1/2 from the centre of enlargement.

Now, we can do this using any of the methods we've learned so far.

But going forward, let's do it by counting squares.

So from the centre of enlargement to the top vertex, the distance is 6.

If we multiply that by our scale factor of 1/2, we get 3.

So our new vertex is 3 away.

From the centre of enlargement to the bottom vertex there, we go down 2 and left 6, multiply those by 1/2, and we go down 1 and left 3.

And finally, to get from a centre of enlargement to this vertex, we go down 4 and left 8.

Multiply those by 1/2, we go down 2 and left 4.

We can now join up our vertices to complete our image, which is half the size of the object.

Let's enlarge this shape again, but this time by a scale factor of 1 1/2 from the centre of enlargement.

We can start in the same way, and see that this vertex is 6 squares from the left, and multiply it by 1 1/2.

We can do that by thinking of it as 1 lot of 6 plus a half of 6, so 6 + 3, which makes 9.

The next vertex is 2 down and 6 to the left.

If we have one of those plus another half of them, we are times them by 1 1/2, which makes 3 down and 9 to the left.

And then finally, the last vertex is 4 down and 8 to the left.

To multiply those by 1 1/2, we've got one lot of those plus another half of them, which makes 6 down and 12 to the left.

Then we can join up our vertices.

Let's check how well we've got on with that.

Izzy is about to enlarge the object by a scale factor of 2 1/2.

The height of the object is 4 centimetres.

What will the height of the image be? Pause, have a go, and press play when you're ready for the answer.

The answer is 10 centimetres.

We can get that by doing 4 times 2 is 8, 4 times 1/2 is 2, so that means 4 times 2 1/2 is 8 plus 2, which is 10 centimetres.

Another question, which diagram has correctly enlarged the object by a scale factor of 1/3? Pause, have a go, and press play when you're ready for the answer.

The answer is B, is in the correct position and the correct size.

The easiest way to see this is probably by looking at the bottom left vertex in the object and image for each of those diagrams. This is because those vertices are directly above the centre of enlargement, which makes more straightforward to check than all the other vertices in those images.

The distance between the centre of enlargement and the bottom left vertex of the object is 6.

Therefore, when we multiply that by a scale factor of 1/3, it means the distance between the centre of enlargement and the bottom left vertex of the image should be 2.

That's what we can see in image B.

It's 2 above, where the other ones are 3 above.

Okay, it's over to you now for Task B.

This task contains two questions.

Here is question one.

It's in two parts.

In part A, enlarge the object by a scale factor of 1/3 from the point marked.

And in B, enlarge the object by a scale factor of 1 1/2 from the point (4,0).

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

And here's question two.

Again, it's in two parts.

In part A, enlarge your object by a scale factor of 1/2 from the point marked.

Now we can see that there's a dotted background rather than a squared background, but we can use the same methods we've learned so far by counting along the dots.

And then B, enlarge the object by a scale factor of 2 1/2 from the point marked.

We can see here that the grid is made out of triangles rather than out of squares.

But we can use the same methods we've learned so far, either by using a ruler if you want to, or counting along the lines and applying our scale factors to those distances.

Pause the video, have a go at these questions, and press play when you wanna see the answers.

Okay, well done everyone.

Here is what our answers look like.

For questions 1A and B, check that your images are the right size, and also in the right positions in relation to the object.

In question one, to check it's the right size, you should have a rectangle that is 2 squares across and 1 square up, because that is 1/3 of the size of the length's 6 squares across and 3 squares up, which is in the object.

And then check that your image is in the right position, it should be 1 square above the centre of enlargement.

In question B, to check your image is the right size, it should be 3 squares across and 3 squares up, because that is 1 1/2 of the size of the lengths in the object, which is 2 squares across and 2 squares up.

And to check it's in the right position, the vertices of your image should be at (4, 12), (7, 12), (7, 9), and (4, 9).

And here is what our answers should look like in question two.

Once again, you need to check the size and positioning of your images In question 2A, the size of the object is 8 units across.

Even though there are no lines on the grid, there are the dots, which means we can start at one vertex and count how many dots, or long it is, to get to the next vertex, which is 8.

Therefore, the size of our image should be 4 units across.

And the position, it should look like it does on the screen now, where you've got 9 dots inside the image, and the bottom middle dot is the centre of enlargement.

Question 2B is made tricky by having the triangle grid rather than a square grid.

But we can still use the same methods.

Start at the centre of enlargement and count how many units we need to travel up to get to the top vertex of the object.

That was 2 units, two little triangles.

Multiply that by 2 1/2 means we need to travel 5 units from the centre of enlargement to the top vertex of the image.

And from there, we can figure out where the other ones are in the original object.

That triangle is 4 units across, and in each direction.

So each of the edges of the image should be 2 1/2 plus 4, should be 10 units across in each direction.

So we have a equilateral triangle that is 10 by 10 by 10 in our image.

Fantastic work today.

Well done.

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

Firstly, we've learned that there are many, many different methods that can be used to enlarge objects.

So we can choose to use whichever method seems most appropriate for the situation that we're in.

Secondly, we've learned some tips for how to check that the images in the correct position in relation to the object and the centre of enlargement.

That is, lines from the centre of enlargement passing through the object's vertices can be drawn to check the images in the correct position.

Because when a line joins a centre of enlargement and a vertex of the object, that line should also intersect the corresponding vertex on the image as well.