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 today's learning outcome is to be able to perform an enlargement and also to describe an enlargement on a given object.

On the screen there are some keywords that I'll be using during the lesson.

You have learned them before in your studies, but you may wish to pause the video here so that you can read them again and make sure you feel confident before we make a start.

Our lesson is on checking and securing understanding of enlargement with positive fractional scale factors.

And we're gonna do this by breaking the lesson into two learning cycles.

The first one is going to be looking at describing an enlargement, and the second is where we will perform an enlargement.

So let's make a start at looking at what we need to describe an enlargement.

So similar shapes are an enlargement of each other and the corresponding edges are in the same ratio.

So if we look at these two similar rectangles here, we can see that the lengths have been multiplied by a half to get from the larger rectangle to the smaller one.

And this is the same if we look at the widths and because that ratio is the same, because that multiplier is the same, then we can say that these two shapes are similar.

If they're similar, then we can also say they are an enlargement.

This is also true for the ratio within the shape.

So if you look at the length and the width within the larger rectangle and the length and the width within the smaller rectangle both of them need to be multiplied by 2/5 to get from the length to the width.

Whereas on these two right angle triangles, the shape is the same, but they're not similar shapes and that means that they are not an enlargement of each other.

If we look between the corresponding edge lengths, the perpendicular edges, the ratio or the multiplier is different.

And if we look within the shape, they are also different.

So because the ratios are not the same, then they are not similar and not an enlargement.

So here's a check, if these two rectangles are an enlargement of each other, then what is the missing length? So pause the video and work out that missing length on the second rectangle, when you're ready to check it, press play.

So it's 30 centimetres, so you could figure that out by looking at the multiplier from 30 to 20 and applying that to the 45.

Or you could do the with using the ratio within the shape.

So Sofia is going to enlarge the object that we can see on the grid by a scale factor of naught 0.

5.

So that's a fractional scale factor.

I've written it with a decimal, but we could write that as a half, but she's not sure where to place the image on the grid.

So we can work out the size of the image, we can multiply all of the edges by a half.

We know that will create a similar shape, a similar triangle, which will be an enlargement.

But where should it be located? Well, this is where the centre enlargement is used.

So this is a point from which an object is enlarged and it is invariant, it doesn't move within the transformation.

So if the scale factor affects the size of the shape, it also affects how far each vertex is away from the centre of enlargement.

So here is the image of Sofia's enlargement of 0.

5, and we can see that from the centre of enlargement to the object, and I'm looking at the top vertex of the object, it's 10 units.

We can use the square background there to measure it.

And the image, the same vertex.

So the corresponding vertex is five units from the centre of enlargement.

And so that distance has also been enlarged by the scale factor of a half.

So here is a check, so Jun has said "The dimensions of A prime, so the image of A are half the corresponding dimensions on A, which is the object.

The distance marked is also half.

That's the two arrows representing distances.

So this enlargement has been completed correctly.

Do you agree with Jun? So pause the video, read what gin has said again, look at the diagram.

Do you agree that this enlargement has been completed correctly? Press play when you're ready to check.

I'm hoping you went for no.

And the reason that it hasn't been enlarged correctly is because those two arrows represent in distances are not for a corresponding pair of vertices.

So if we look at the shorter arrow, that's from the centre of enlargement to the bottom right vertex of the image, whereas the longer one is from the centre of enlargement to the bottom left vertex of the object.

So we must make sure that when we're looking at distances that they are the corresponding distances.

The centre enlargement can be anywhere on or off of the object.

It can be in various positions.

So it might be outside of the object, it might be on an edge of the object, it might be on a vertex or it could even be inside.

And the centre of enlargement is where with alongside with the scale factor is where we get the position of the image.

And we need to be able to locate that centre of enlargement when we look to describe a transformation.

And we can find it by using lines that pass through the corresponding vertices.

So if we look at this example here, we've got a pair of similar triangles and therefore we know an enlargement has taken place.

The reason the object is in a specific position is because of both a scale factor but also the position of the centre.

So where is that centre? So if we draw a line that passes through these two corresponding vertices, one's on the object and one's on the image and do the same for another pair of corresponding vertices, and finally the last pair, you can note that they all meet at this particular point and that particular point, that coordinate of negative nine four is the centre of enlargement.

And so we can use these lines to locate the centre enlargement when we look to describe.

So if we want to write a full description, then we need to state the transformation, which in this case is an enlargement.

We need to state the centre of enlargement, which is a coordinate here, and also the scale factor of the enlargement.

So the object and image are not congruent.

So they're not exactly the same size, but they are similar, which means that we know it's an enlargement.

So that's identifying the transformation that's taken place.

So at the bottom you can see the start of my description, the object has been enlarged.

And then we need our scale factor.

So our scale factor we can calculate by looking at the multiplier between corresponding lengths.

I'm using the vertical height, perpendicular height of the two triangles.

So our object has a height of four and our image has a height of 10.

So the scale factor is 10/4, which simplifies to 5/2, which is 2.

5 as a decimal.

So it is a fractional scale factor.

So our description is now extended to say the object has been enlarged by a scale factor of 2.

5.

The last thing we need to give is that centre enlargement and our lines have located the centre negative nine, four.

So our full description would be, the object has been enlarged by a scale factor of 2.

5, which you could write in fraction form if you wanted, or a mixed number, from the point negative nine, four.

That is a full description of an enlargement.

So to check your understanding of that, there is some missing parts to the description here.

So you need to fill in the blanks.

So pause the video, decide what they should say, and then press play to check yourself.

So the first missing part was the word enlarge.

Remember you need to state the type of transformation.

So this we can see are not congruent, the object and the image, but they are similar, so that's why it's an enlargement.

Now the scale factor, so we need to look at a corresponding pair of distances, that could be a length or a distance from the centre to a vertex.

I'm going to use the lengths.

So the shape A, which is the object has a horizontal maximum distance of three, whereas the image is seven.

So the scale factor would be 7/3.

If you've written that as two and 1/3, which is a mixed number, that's perfectly fine, they're equivalent.

And then lastly, we need to state the point, the centre of enlargement.

And we can see our ray lines all intersect at negative two, one.

So our full is shape A has been enlarged by a scale factor of 7/3 from the point negative two one.

So onto the first task of this lesson on question one, you need to state the enlargement scale factor for each pair of similar shapes.

So pause the video whilst you do that, and then when you press play, we'll move to question two.

So here's question two, on question two you need to locate the centre of enlargement for each object and image.

So they're labelled A and a prime, et cetera.

So you need to locate the centre of enlargement.

So pause the video whilst you work your way through those four pairs of object and image.

And then when you press play, we'll move to question three.

So here's question three, which is the last question on task A.

You need to fully describe the single transformation that has taken place.

For part A, it's looking at the image and the object that are both labelled A and A prime.

And for B, it's looking at the pair of similar shapes, B and B prime.

So pause the video and when you're finished and press play, we'll go through the answers to all of these questions.

So question one, you needed to state the enlargement scale factor.

So choose a pair of corresponding lengths or edges in this case and find the multiplier.

So on part A, it was 1/3, on part B, it was 1/2.

And on part C, it was also 1/2.

Part C, there wasn't any edges that were laying directly on a horizontal grid line or a vertical grid line.

So you needed to have a different method.

So one method would be to look at two vertices within the shape and look at how they were horizontally and vertically different to each other.

An alternative way is to look at a distance across the shape.

So if we look at the perpendicular height of the object, the perpendicular height of that triangle is four squares.

And therefore if we look at the same perpendicular height on the image, it is two squares.

We can see that the distance, that length has been multiplied by a half.

On question two, you needed to locate the centres, and this was using the ray line.

So remember you're gonna draw a line that passes through the corresponding vertices.

They will all intersect at the centre.

So four A onto A prime, the centre was negative four, 10.

on B to B prime, the centre was 10, three.

On C to C prime, the centre was negative six, negative four.

And lastly the centre for D to D prime was 15, negative four.

And here you can see examples of centres of enlargements that are outside of the object, centres of enlargements that are inside and centres on a vertex.

Lastly, for question three, you need to bring all of that together.

So coming up with a scale factor, finding centres to be able to write a full description.

So for part A, object A has been enlarged by a scale factor of 3/2 two or 1.

5 if you wrote it as a decimal, from the point negative one, negative five.

And then for part B, object B has been enlarged by a scale factor of 0.

5.

Once again, you may have written that as a fraction, so half from the point negative five, three.

It is important that you have said that the shape has been enlarged even if the image is smaller than the object.

We are now up to the second learning cycle and this learning cycle, we are actually going to perform some enlargements.

So ray lines from the centre of enlargement go through the corresponding vertices on the object and the image.

And this is why we can use them to locate the centre.

We can use this element to perform an enlargement and get the size and the location correct depending on the scale factor.

So if we have a look at this diagram here, we've got a centre of enlargement marked at four two, and then the object, which is a trapezium.

We've drawn the ray line that starts at the centre and passes through each of the vertices of that trapezium.

Because a trapezium is a quadrilateral, we've got four ray lines, depending on the scale factor depends on the size and the location.

But what we do know about where it will be located, the image that is, is that the vertices will also be on those ray lines because remember, those ray lines pass through the corresponding vertex on the object and the image.

So if the scale factor was this, this would be where it's located and the size of it, but it could be here, it could be here.

All of these would be images for that object depending on the scale factor.

So how do we use ray lines and a ruler to help us perform an enlargement? So if we haven't got a grid, we've just got a blank sheet of paper and we want to enlarge this object, which is a triangle by a scale factor of a half from the mark centre of enlargement.

How do we do this? Well, we're gonna make use of those ray lines.

So the first thing we're gonna do is we're gonna draw those ray lines starting at the centre of enlargement and passing through the three vertices of our object.

Next, we are going to measure with our ruler because we know that the distances is what changes when we enlarge, we multiply by the scale factor.

So if I measure from the centre to that vertex, it's 5.

9 centimetres.

So I need to enlarge that distance by the scale factor half.

If I multiply that distance by 1/2, we get 2.

95.

So now if I measure from the centre, it's important that we're always measuring from the centre, 2.

95 centimetres along the corresponding ray line, I can locate the image of the vertex.

We repeat this for all of the other vertices.

For this it's two more because it's a triangle.

So if I measure this, it's 8.

5.

If I multiply it by the scale factor, I need to mark 4.

25 centimetres along that ray line from the centre.

And that would be the location of the image for that vertex.

And then lastly, measuring this is 4.

9 centimetres, multiply it by 1/2 is 2.

45 centimetres, mark it.

And then the last step is to draw the line segments between the vertices to form the image.

So our image is in the correct location, which is exactly half the distance from the centre as the object.

If however, the object was on a grid background, then another method is to count the grid rather than use the ruler.

So we're still measuring a distance that we can then scale or using our multiplier.

However, we are using counting instead of a measure.

So if we want to enlarge this object by a scale factor of 1/3 from the point negative three, negative two, how do we do this? Well, first of all, we're gonna plot our centre of enlargement, which is already there.

We're then gonna choose a vertex.

It doesn't matter which vertex you start on.

So I've chosen the bottom right vertex, and I'm gonna count how many squares using the grid to get from the centre of enlargement to that vertex.

There is no horizontal displacement, it's directly above.

So it's six squares up.

We are now going to multiply that distance, six squares up by the scale factor.

And that gives us two because six multiplied by 1/3.

The scale factor in this example is two.

So we now know that the corresponding image of that vertex is two squares above the centre.

And I'm gonna mark that position.

We're gonna repeat this now for all the other vertices.

So the top vertex of the triangle on the object is three to the left.

So using the grid background, three to the left, three squares that is and 12 squares up.

So I'm going to multiply three squares to the left, 12 squares up by 1/3, the scale factor, and that gives me one square to the left and four squares up.

I can mark my location.

Final vertex is three to the left and six up to the object, multiply it by the scale factor, so one square to the left and two squares up for the image.

Once I've plotted my three corresponding vertices for the image, we form our image using line segments.

So here's a check, the object is enlarged from three, one.

So that's our centre of enlargement by a scale factor of 1/2.

So 0.

5, which point, so either A, B, C, or D shows where the image of A is located? So A is a vertex on the object.

And where would the image of A be located? Pause the video and when you're ready to check press play.

So the image would be at the position marked with a D.

And again, we're gonna count from the centre to that vertex.

So from the centre to vertex A, which is on the object, it's two squares left and six squares up.

So we're gonna multiply by the scale factor, which means would be one square left and three squares up from the centre.

And that is the position of D.

Ray lines can be used when we don't have a grid background and we need to measure the distance between a vertex and the centre direct distance using that line.

But another thing we can use the ray lines for is to check our working.

So to check the image is in the correct position.

So all the vertices of the image should be on the ray line with the corresponding vertex.

So this one here, this image would be in the correct position because the corresponding vertex to the object is on the same ray line and all of them are on the ray line.

Whereas on this one the image is too low.

It might be the correct size, but its position is not correct.

And you can see that those three vertices are not on a ray line.

And lastly, it might be that you've done most of the enlargement correct, but one bit's gone slightly array.

And we can use our ray lines and see that here we've got one vertex that is wrong.

So two of the vertices are on the corresponding ray lines as it should be.

And that third one is the one that's in the wrong position.

By that being in the wrong position, our object and our image are no longer similar.

So it isn't an enlargement, but it would highlight to you that you'd need to go and check that part of your image.

So which of the following diagrams shows a correct enlargement? Pause the video, have a look, and then when you're ready to check, press play.

So the answer is B, on A and C, we do have some of the vertices on the image lying on a ray line, but not all of them.

Whereas on B, we can see that all four vertices are on a ray line.

So we're up to the last task of the lesson for you to do some performing of enlargements.

So question one A, you need to enlarge object A from the origin by a scale factor of a 1/3.

Remember you can use ray lines to check your solution.

And then on B, you need to enlarge object B from the 0.

10.

Make sure you plot that in the correct position by a scale factor of one and 1/2.

So that's been given as a mixed number.

That's obviously equivalent to a improper fraction of 3/2, it could be written in decimal form as 1.

5.

So pause the video and complete those enlargements.

And when you're ready for question two, press play.

So here we have question two, which is the last question for this task.

And again, you're gonna complete the transformations as described.

So part A is enlarge object A from the 0.

65 by a scale factor of a half, and for part B, enlarge object B from the 0.

33 by a scale factor of 3/4.

So pause the video.

When you finish doing those and you press play, we'll go through our answers to both questions.

So on question one, we can see the answers here.

So you can see the image has now appeared.

You can check using ray lines.

As I said, if your ray lines do not all intersect back at your centre of enlargement, then your image is in the wrong location.

Or it may be that it's the wrong size.

You might want to pause the video and just check vertex by vertex from the image on the diagram here and your image to check you've got the location correct.

Moving on to question two, once again, you can check using ray lines or pause the video and check the vertices of the image.

On both of these we can see that the image and the object touch or that they overlap.

And that's because of where the centre is located, especially on B, but also because of the scale factor.

So remembering that it's okay if your object and image overlap or touch, they don't have to be separate from each other.

So to summarise today's lesson, which was check in and secure an understanding of enlargement with positive fractional scale factors or enlargement is a type of transformation.

And if we wish to enlarge an object, then you need to know the scale factor and the centre of enlargement because that determines where the image is located and the size of it.

All lengths are multiplied by the scale factor.

So that includes both the object edges and the distance from the centre of enlargement.

If we need to describe an enlargement, then we need to state that the object has been enlarged.

Often that bit is overlooked, but you must make sure you write down that it is an enlargement.

Then you need to give the centre of enlargement, which is often a coordinate.

And the scale factor, ray lines are a really useful thing to use when we work with enlargements, both to locate the centre of enlargement, but also to check the position of your image if you have performed that in enlargement, it's a really clear way of seeing that you've got it correct.

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