Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

I really hope you're ready to try your best and ready to learn.

So our lesson outcome today is to be able to perform an enlargement on a given object.

So there are some key words that I'll be using during the lesson that you have met before in your studies.

So you may wish to pause the video just to refamiliarize yourself before we get started, but I'll leave that up to you.

So our lesson is on enlargement using a negative scale factor and we're gonna break that into two learning cycles.

In the first learning cycle, we're gonna focus purely on negative integer scale factors.

And then when we move on to the second learning cycle, we'll look at negative fractional scale factors.

So let's make a start at enlarging using that negative integer scale factor.

So to enlarge this object by a scale factor of two, all distances need to be multiplied by the scale factor.

So I've drawn lines, ray lines from the centre of enlargement through each vertex and I'm gonna use a ratio table.

If I measure the distance from the centre of enlargement to a vertex on that object is 2.

8 and we can say that that is our unit length, so our one.

if we want to scale that up to a scale factor of two, then on the scaling side, that will now become two.

That unit length is now gonna be twice as long.

So we can use our ratio table and look at the ratio between all of those values to figure out and to work out what that distance from the centre of enlargement to the image of the vertex will be and that's 5.

6.

So from measuring from the centre of enlargement up 5.

6 along the ray line, I can locate the image, the corresponding vertex.

We can do the same thing for the next vertex on the object, which happens to be 2.

8.

As a distance, we know that 2.

8 is our unit length and so, when we scale that, we end up with a distance of 5.

6.

So we now know the location of the second vertex on our image.

Our third vertex, if we measure that is 4.

5.

So this is now gonna become our unit length.

If we scale it using our scale factor, then it needs to be nine away and we can measure along the ray line from the centre of enlargement and mark where nine would be.

We can then join those up with line segments to form the image.

So here is an enlargement by a scale factor of positive two where all distances has been multiplied by two.

So what I'd like you to do for this check is to fill in the missing value in each of the ratio tables.

So press pause and figure out those missing values.

When you're ready to check your answers, press play.

So the first one, the missing value would be nine.

We've highlighted already on the ratio table, the sort of multipliers either horizontally or vertically that could help you.

The scale factor would be three here.

On the next one, the missing value would be two.

The scale factor is two.

If the distance was 3.

5 and we need to multiply by two, then that becomes seven.

The last one, the missing value is 2.

5.

So after scaling, we can see that the distance is 10 and the scale factor was four.

So we can divide by four to get the initial distance from the centre of enlargement to a vertex on the object.

So we want to think about negative scale factors, and actually, if the scale factor is negative, the same process takes place.

So if we've got our ratio table here, I've measured from the centre of enlargement to a vertex on the object, while we're going to scale to a scale factor of negative two this time.

So if from one to negative two is multiplying by negative two, then we do the same thing to the distance.

And when we do that, we get negative 5.

6.

But what does that actually mean? Well, we can't have a negative distance.

The negative tells us the direction has changed.

So the distance from the centre of enlargement is going to be 5.

6, but in the direction away from the object.

So the negative is changing the direction.

So here is the location of the image vertex.

We can repeat that on all of the other vertices.

So once again, this distance is 2.

8, so we need to measure 5.

6 in the opposite direction away from the object but measuring from the centre.

And again, the last point is 4.

5.

So we would measure nine away in the opposite direction along the line and we can join those up to form our image.

So what do you notice about the image here? Have a moment to think about that.

How is that different to a positive scale factor enlargement? Well, I'm hoping that you notice that it is twice as size, although lengths of the object have been doubled, but it's a rotation of 180 degrees.

So it has increased in size, but it's also rotated by 180 degrees.

So which of the following shows an enlargement by a negative scale factor? Pause the video and look at those diagrams and then when you're ready to check your choice, press play.

So it is C.

So if a negative scale factor has been used, then the object and the image will be rotations of 180 degrees of each other.

So we can see on A that that would be a positive scale factor.

On B, that isn't actually an enlargement.

And C, we have a negative scale factor.

So if the enlargement was on a grid, then the enlargement can happen by counting horizontal and vertical distances.

So rather than using a ruler to measure that direct distance between the centre of enlargement and a point, we could use the grid background instead.

So if we wanted to enlarge this object, which is a triangle by a scale factor of negative two from the 0.

67, how do we go about doing this? Well, firstly, we plot that centre of enlargement.

So we plot the coordinate, making sure we get that in the right position.

Then we're gonna count the number of squares both horizontally and vertically if necessary from the centre of enlargement to a vertex on the object, it doesn't matter which one we choose.

So to count to the bottom left, it's one square left.

So I'm gonna call that negative one because I've moved one X value down the line and then we are moving up one, the wide coordinate has increased by one.

So if I now scale that, if I multiply that by the scale factor, which is negative two, then our distances become positive two and negative two horizontally and vertically.

So positive two is a movement to the right.

Remember that that negative scale factor, the negative part of the distance is a change of direction.

So if we'd previously gone left, we've now gone to the right and then if we previously went up, we are now going down and the distance has multiplied by the scale factor.

So that is the location of the vertex image.

And we repeat this for all of the vertices on the object.

So this top vertex is three up from the centre, which means that the image of that vertex will be six down.

The negative two is changing the direction and doubling the distance.

The last vertex is two to the right and one up from the centre.

So where would that final vertex be? Well, we're gonna multiply by negative two, which means it'll be negative four or four to the left and be negative two or two down.

Once we've got the location of the three vertices, because it's a triangle, join them up and form the image.

So this is the image and we can see that it is twice the size and a rotation of 180 degrees.

So for your check here, can you fill the blanks, how many squares and in what direction? Press pause and then when you're ready to check, press play.

So the horizontal direction will be six squares left.

And that's because the vertex A is two squares right to the object.

So when you multiply by negative three, the direction changes, the right will become a left and two times by three is six.

And finally, what about vertically? Well, it'll be three squares down.

That's because the object, the vertex A on the object is one square up from the centre of enlargement.

So it needs to be three squares down from the centre of enlargement for the image.

So we're now up to the first task of the lesson where you're going to be performing some enlargements using negative integer scale factors.

So in question one, you've got two enlargements to perform.

So read it carefully, make sure you plot the centre in the correct place.

And then counting and multiplying by the scale factor.

Press pause whilst you work through those two parts and when you press play, you've got question two.

So here's question two.

Question two is much the same.

Different places where you're gonna put the centre and obviously, the object on A and B is slightly different.

So press pause whilst you're performing those and when the press play, we'll look at the answers.

So here's the answers to question one, part A and part B.

Both are rotations by 180 degrees because of the negative scale factor and both of them are enlargements multiplied by two and multiplied by three with that negative change in the direction that you have gone, it may be that you wish to draw ray lines and check that you have got these in the correct position or pause the video and check the coordinates of each vertex on the image to make sure that you have it in the correct location.

And then we move on to question two.

Question two, once again, you were multiplying by a scale factor that was negative and so therefore, the image will be a rotation of 180 degrees compared to the object.

This time our centre of enlargements one was a vertex on part A.

And so, you can see they've got a shared vertex.

And on part B, the centre of enlargement was within the object.

So there is an overlap between the two.

So our second learning cycle is looking at enlarging with negative scale factors, but more importantly now they are gonna be fractional.

So if we look at the diagram here, we've got the image, which is the purple one and its object.

And the image of an enlargement where the scale factor is negative one is going to be the same distance away from the centre of enlargement and congruent to the object.

The only difference is it is rotated 180 degrees.

So because the scale factor has an absolute value of one, then the distances will be identical.

If you multiply by one, it doesn't change the value.

The negative is where we find this rotation of 180 degrees because you are going in the opposite direction.

So it's the negative element of the scale factor that causes that rotation by 180 degrees and it's the absolute value of the scale factor that determines the size of the image.

So the absolute value of negative one is one, and that's why the size of those two, the object and the image are the same, which makes them congruent, which means that the image from an enlargement of scale factor negative two is twice the size of both the object and also, an image of scale factor negative one.

And that's because as we've already seen a scale factor negative one produces a congruent shape.

If we now look to think about fractional scale factors, well, fractional scale factors can make the image smaller than the object.

So if the scale factor was negative a half, then we know it'd be rotated 180 degrees because of the negative.

And the absolute value being 1/2 means all of the distances have been divided by two, the image is now smaller.

If it was negative a quarter, it would be even smaller still.

However, they can make the images larger too, so if the absolute value is greater than one.

For example, negative three halves, which is the same as negative 1.

5, the negative has caused the rotation by 180 degrees because you are measuring in the opposite direction each time.

The 1.

5 element of the scale factor is what is increasing the size of the object.

And negative 12/5 would be even larger still.

So here's a quick check, true or false, and you need to justify your answer.

So pause the video, read through it, make your decisions, and when you're ready to check, press play.

So it's false, that a fractional scale factor makes the image smaller.

So the reason is it depends on the absolute value of the scale factor.

If we look at performing an enlargement using a negative fractional scale factor, then we can use the same methods that we've previously used when we've looked at integer scale factors, whether they were positive or negative.

So here, if we look at this example where we want to enlarge object A by a scale factor of negative 1/3 from the point marked.

So the point marked is our centre of enlargement.

So again, if we bring back our ratio table because we haven't got any grids, et cetera, then we are going to use our lines, go from the centre of enlargement to a vertex and measure it, measure it using a ruler, and this comes out as 3.

3 units.

That's gonna be our unit length.

We then need to scale it by negative a third, so multiplying by negative a third in the vertical direction on the ratio table, we can do the same to our distance, which gives us negative 1.

1.

What does that mean? It means we need a distance of 1.

1 in the opposite direction from the centre of enlargement.

So we can mark that on the line.

If there were to be a grid, then we can use the number of squares instead.

So rather than you measuring the direct distance from the centre of enlargement to a vertex on the object, we can do the counting.

So let's continue through this enlargement using counting.

So if we choose a vertex, so I've chosen this vertex here, it is three squares to the right of the centre of enlargement.

So when I scale that by a scale factor of negative a third, then the right will become left because the negative changes the direction.

And when we scale it by third, then we become one.

So that is the position for the image of that vertex.

We can repeat that for the other two.

So this one is nine squares right from the centre.

So therefore, the image will be three squares left multiplying by negative a third, that's making the distance shorter by multiplying by a third.

And it's changing the direction because it is negative a third.

And the last one is 7.

5 squares right, and three squares up from the centre of enlargement.

Remember, you're always counting from the centre of enlargement, multiplying that by negative a third means that we need to go 2.

5 squares left and one square down.

Once we've plotted the vertices, we can join them up to form our image.

So here we can see it's a trapezium rotated 180 degrees and a third of the size.

Importantly though, it's also a third of the distance from the centre of enlargement.

We can use ray lines to check the size and the position are correct.

So a line that passes through the corresponding vertices where they will all meet, they'll all intersect at the centre of enlargement.

So here is a check, it's quite a worded check, so I'm gonna read it to you.

You can pause the video and reread it to yourself before you make your choice.

A vertex on the object is six squares right and three squares down from the centre of enlargement.

If the scale factor is negative 1.

5, where is the corresponding vertex relative to the centre of enlargement? Is it 4.

5 squares left to 1.

5 squares up? Is it nine squares right and 4.

5 squares down? Or is it nine squares left and 4.

5 squares up? So pause the video.

As I say, you might wanna read that again, process it and then make a decision on A, B or C.

Press play when you're ready to check.

So this would be C.

So multiplying those counting distances by negative 1.

5 will change the direction, that's what the negative part is changing the direction and also will change the value.

So six multiplied by negative 1.

5 is negative nine.

If the direction was right, it's now gonna go left.

So it's nine squares left and similarly it's 4.

5 squares up.

So we're now on the final task of the lesson where you're gonna perform some more enlargements.

This time they are negative fractional scale factors.

So here is question one.

You've got two parts, two enlargements to do.

So pause the video whilst you perform those, and then when you press play, we've got question two as well.

So question two is now on the screen.

Again, two enlargements for you to perform.

Make sure you are plotting the centre of enlargement in the correct place.

You can use your ray lines to check your answer as well and counting and multiplying by the scale factor.

So pause the video and then when you're ready for the answers to task B, press play.

So the first part of question one is on the screen.

So the image of A, you can see that it is smaller because the scale factor is negative a third, and it is also a rotation by 180 degrees because it is a negative scale factor.

And then here is now the image of B.

The image of B actually nests neatly into the L shape of the image of A to create a rectangle.

You may wish to pause the video to check the vertices against your own solution to check that you're okay with that.

It's hard to tell that a rotation of 180 degrees has taken place with a rectangle.

So make sure you have got that in the right location.

Finally, here's question two.

Again, the two solutions or the two images are on the screen.

The centres of enlargement were a vertex and also within.

So we have got again, a vertex I shared for the image of A and A.

And then also the image of B is overlapping the object B.

So mind be mindful, take your time to check against your own one.

A is another shape that is hard to tell that a rotation has taken place by 180 degrees, whereas on B, we can see that a little bit more clearly because of the nature of the shape.

So we're up to the summary of the lesson, which was enlarging using a negative scale factor.

So enlargement is a transformation which produces a similar shape to the object.

So it's a similar shape if you end up with two shapes that are not similar, then your enlargement has gone wrong, unfortunately.

So to enlarge an object by a negative scale factor, all the distances are to be multiplied by the scale factor.

So actually, the process is no different to using a positive scale factor.

The difference is the sort of outcome.

So the object and image will be rotations by 180 degrees of each other in an enlargement by a negative scale factor as well as the change of size.

So that is where the negative scale factor has an effect.

The change of size is because of the absolute value, so it can get smaller and it also can get larger.

But if the scale factor is negative, then the image and the object are rotations of 180 degrees of each other.

Really well done today and I look forward to working with you again in the future.