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 on a given object and also to describe an enlargement that has taken place on a given object.

There are some key words that will be usin' during the lesson, which you have learned before, but you may wish to pause the video so that you can refamiliarize yourself before we make a start.

So the lesson is on checkin' and securin' understanding of enlargement with positive integer scale factors.

And we're gonna do this by breaking the lesson into two learning cycles where the first one is describing an enlargement and the second part is performing an enlargement.

So let's make a start on describing an enlargement.

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

So we could look at this as a ratio table or just the link between the two corresponding edges.

So the five centimetre on the smaller rectangle corresponds with the 10 centimetre length on the larger one.

And so we can see that five multiplied by two gives us 10.

And because they are similar, this works on the widths as well.

And so the multiplier of two is the same between the shapes.

So the corresponding edges are in the same ratio and this is the same within the shape as well.

So if you look, the length and the width are both multiplied by two fifths and you can see that on the ratio table.

These two shapes, however, are not an enlargement of each other.

So they're both right angle triangles, but they are not similar.

And that's because if we look at the corresponding edges, there is not the same ratio, there is not the same multiplier between them.

So the four needs to be multiplied by two to become eight, whereas the three needs to multiply by five thirds to become five.

And similarly this would mean that the ratio of the sides within the shape are not equal either.

So you can see that as well with times in by three quarters and times in by five eighths.

So if the two shapes are not similar then an enlargement cannot have taken place.

So here's a quick check.

This is an enlargement.

These two rectangles are similar.

So what is the missing length? Pause the video whilst you work that out.

And when you're ready to check your answer, press play.

So the missing length was three.

So Aisha is going to enlarge the object by a scale factor of two, but she's not sure where to place the image on the grid.

So we have an object and we want to enlarge it by a scale factor of two.

So we know that the image will be similar to this object, but where should it be located? So the centre of enlargement is a point from which an object is to be enlarged, and this is where we can the position that the image needs to be in.

the scale factor not only affects the lengths of the image, but also how far each vertex is away from the centre of enlargement.

So if we look at the vertex at the top of the object, that is five units from the centre of enlargement.

And if we look at the corresponding vertex on the image, that is 10 units from the centre of enlargement, because the scale factor is two.

So the distance between a vertex and the centre has also been multiplied by the scale factor.

The position of that centre of enlargement defines where the image will be placed in relation to the object.

So here we've got the centre of enlargement to the left five units away from the top vertex of the object.

And because the scale factor is two, the image is 10 units away from the centre of enlargement.

If it changes, if the centre enlargement moves to a different position, then because the distance has changed, then the image will be in a different position as well.

In each of these, the image is still the same size because the scale factor is two.

What is different is the position that it's located.

So the centre of enlargement can be found by drawing lines through those corresponding vertices of the object and the image.

So here would be one line that passes through the corresponding vertices, another line, and the third line.

Because it's a triangle, we only need the three.

And they all intersect at the centre of enlargement.

So this is the method to find the centre of enlargement when we're trying to describe what has gone on in the transformation.

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

And that point will remain in variant in the image.

So it doesn't move.

The centre of enlargement is in the same position for the object and the image.

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

And so you can see that the image is in a different position dependin' on where the centre of enlargement is.

So we are looking to describe an enlargement, and a full description of the enlargement requires both the scale factor as well as the centre of enlargement.

So if we look at this example where we want to fully describe the transformation that maps A, the rectangle onto the image of A, A prime.

So firstly, we need to think about the size and the size will help us with the scale factor.

So the image has a width of three units and the object had a width of one.

So what was the multiplier? Well, the multiplier would've been three.

One multiplied by three gives us the three on the image.

Now we need to find that centre of enlargement and we're gonna do that by drawing the lines passing through the corresponding vertices and they intersect at the coordinate 11, 4.

So we now have the centre of enlargement as well as the scale factor.

So our description would be object A has been enlarged.

It's really important that we mention that the transformation is an enlargement by a scale factor of three from the point 11, 4.

So that is our full description of the transformation that mapped shape A onto shape A prime.

So can you have a go at that for a check? Write a full description of this enlargement.

Pause the video and then when you're ready to check, press play.

So you should have written something similar to object A has been enlarged.

It's really important that you've mentioned it's an enlargement.

The scale factor was two.

So you could find the scale factor by comparing corresponding edges.

Remember that an enlargement creates similar shapes and then from the point -2, 1 and the ray lines met at the centre of enlargement.

So onto the first task of the lesson where on question one need to state the enlargement scale factor.

So by using corresponding edges, work out the multiplier that creates these similar shapes.

Pause the video whilst you're going through parts A, B, and C.

And when you're ready to move on on task A, press play.

So on question two, you are locating the centre of enlargement.

So we've got pairs of similar shapes, they have been enlarged, but where was the centre of enlargement? So using ray lines passing through the corresponding vertices, you should be able to locate it and write down the pair of coordinates.

So pause the video whilst you locate the centres.

And then when you're ready for the next question, press play.

So on question three, part of the descriptions are missing.

So you need to fill in the blanks.

So use the diagram, you may need to draw the lines on to find the centre of enlargement, especially for part B.

So press pause and when you're ready to move to question four, press play.

So here we have question four where you need to fully describe the single transformation that has taken place from A onto the image of A, and from B onto the image of B.

So press pause and then when you're ready for the answers to all of task A, press play.

So on question one, you needed to state the scale factor in each case.

So that was by comparing the corresponding edges.

So on part A, if you looked at the top edge on both the image and the object, the top edge of the object was three and it was six on the image.

So it is multiplied by two.

So the scale factor would be two.

For part B, again, if we use the top edge on the object, it's one unit and on the image, it's three.

So it's been multiplied by three and that means the scale factor is three.

And lastly on C, the scale factor is two.

This one's slightly different because none of the edges are along a horizontal or a vertical on the grid.

So you may need to look at both movements in horizontal and vertical directions.

So if you went from the top vertex of the object to the right hand vertex on the object, that is two squares to the right and one square down.

If you do that on the corresponding vertices of the image, it's four squares to the right and two squares down.

So it is twice as far, it has been doubled and that's why the scale factor is two.

On question two, you needed to locate the centre of enlargement.

So this is where we use our ray lines passing through corresponding vertices and they should all intersect at that centre.

So on part A, they intersect at -1, 8.

So there, we can see that the centre of enlargement is outside of the object.

And on part B, they intersect at -7, -4.

The centre of enlargement is within, is inside of the object.

Question three, you needed to complete the descriptions.

On part A, you are missing the word enlarged.

So making sure that we state the type of transformation and also the scale factor.

So using the corresponding edges to see that it had been enlarged by a scale factor of two.

And then on part B, you needed to locate the centre of enlargement, which was at -4, -5.

And then finally on question four, you needed to write the full description for both A onto A prime and B onto B prime.

So for part A, object A has been enlarged by a scale factor of three from the point 3, 8.

And for part B, object B has been enlarged by a scale factor of two from the point 1, 7.

So we're now up to the second learning cycle where we are going to perform an enlargement as opposed to describe it.

So ray lines from the centre of an enlargement go through the corresponding vertices on the object and the image and we use them to locate the centre of enlargement previously.

This element can actually be used to perform an enlargement as well.

And this allows us to get the size and the location correct depending on the scale factor.

So on this diagram, you can see the object, a trapezium, and the centre of enlargement has been marked at 2, 1.

If we draw our ray lines on, so starting at the centre of enlargement and passing through each vertex, so we've got four ray lines because it is a quadrilateral, then depending on the scale factor, depends on where the image will be located.

But what we do know is that all of the vertices of the image will also be on a ray line.

So if the scale factor was two, it would be positioned here because the distance from the centre of enlargement to the object has been doubled to get this distance from the centre to the image.

And you can that all four vertices of the image are on each of the ray lines.

If it was three, it would be positioned here.

If it was four, it'd be positioned here.

And if it was five, it would be positioned here.

So the scale factor has enlarged the shape in terms of its lengths, but also the distance from the centre of enlargement.

But it's in that position because of the ray lines.

So how do we enlarge, how do we perform an enlargement using a ruler and those ray lines? So if we look at this example where I've got an object, another trapezium, I want to enlarge it by a scale factor of two from the marked centre of enlargement.

So here, we are not on any grid background, so we're gonna draw our ray lines using our ruler.

Remember that they will go through starting at the centre of enlargement and through each of the vertices, then we're gonna need to measure and because it's the distance that is going to be multiplied by the scale factor.

So we're gonna measure from the centre to the vertex.

I've chosen this particular vertex, but we could've started on any of the four.

I'm measuring along that ray line, that direct distance between the two and it measures as 2.

5.

So now I'm gonna scale up that distance using my scale factor.

So 2.

5 times by two is five.

So the image of that vertex, the corresponding vertex will be five centimetres away from the centre along that ray line.

And so I can mark the position of the vertex.

We are now gonna repeat this process for all of the other vertices along their corresponding ray line.

So I can measure this distance, it's 1.

7, the scale factor is two.

So I multiply it, which is 3.

4, measure that from the centre along the ray line and mark the position of the vertex.

Again, I can repeat this, 2.

55, pretty accurate measurement there.

Double it, 5.

1, and mark it, and last vertex of the quadrilateral, which happens to be a trapezium.

Measure it, double it, mark it, join them up.

And there is our image.

So we've used the ray lines to ensure we get the image in the correct position.

So here's a quick check on that.

Once you've multiplied the distance from the centre to the vertex of the object, you measure that distance from the vertex of the object to find the image's vertex, true or false? Pause the video, probably want to read that through again and process what that's actually saying.

And when you're ready to move on to the justification, press play.

So that is false.

So why is that false? Is it because all length are multiplied by the scale factor in an enlargement or is it because the distance from the centre of enlargement is multiplied so it should be measured from the centre of enlargement? Pause the video and think about the justification to why the first statement was false.

When you're ready to check, press play.

So it was B, it is true that all lengths are multiplied by the scale factor in an enlargement, but the reason the first statement was false was because it was suggesting that you should measure from the vertex on the object, but you should always be measuring back from the centre of enlargement.

So if the object was on a grid background, then another method is to count the grid rather than using a ruler.

So we're still looking at the measurement, but instead of using a ruler to get a measurement, we are gonna use the grid instead.

So here is an example of that.

If we want to enlarge the object by a scale factor of three from the point 3, -2, so our centre of enlargement is 3, -2 and there is our object, then the first thing would be to choose a vertex.

Doesn't matter which one you choose, but we choose a vertex to start with.

We're then gonna count.

And this is the difference from using our ruler to measure the distance, we are gonna use the grid to count.

So it's one to the left, one unit or one square to the left and two units, two squares up.

So when we multiply that by the scale factor, which remember in this example is three, instead of going one square to the left, we will go three squares to the left.

Instead of going two squares up, we will go six squares up.

And so then we can mark the corresponding vertex.

We are now gonna repeat this for all the other vertices.

So this vertex is two to the right two up.

So the image will be six to the right six up.

This vertex is one to the right three up.

So its image is gonna be three to the right nine up.

And then lastly, the final vertex on the object is one to the left to three up.

So its image will be three to the left nine up.

Once we've marked all four vertices, then we're gonna join them up to form our image.

So here is a check for you.

If the object is enlarged from -2, 7 by a scale factor of two, which point, so either A, B, C, or D shows where the image of A is located.

So A is a vertex on that object and where would its image be located? Pause the video and then when you're ready to check, press play.

So it would be located at B.

So if we count from the centre enlargement to vertex A, it's three to the right and one down.

The scale factor for this enlargement is two.

So if we double that, that would be six to the right and two down from the centre of enlargement, which lands on B.

So ray lines are a really useful way of checking that you have positioned your image correctly.

And because all of the vertices of the image should be on a ray line, they should be on the same ray line as its corresponding vertex on the object.

So here, we would see that this is in the correct position.

If the object is the purple smaller triangle, its image, all three vertices, the corresponding vertex is on the corresponding ray line.

Whereas this one, the image although is the correct size, is not in the correct position because none of the vertices appear on the ray line like they should.

And lastly, it may be that you've done most of it correct, but you get one wrong.

And then you can see that with the ray lines that the ray lines pass through two of the vertices but not the third as you could then go and check that position of that third vertex.

So which of the following shows a correct enlargement from the marked centre of enlargement? So pause the video, have a look at each diagram and which one do you know is in the correct position because of where it is with the ray lines? Press play when you are ready to check if you are correct.

So A is the correct one.

A is correct because all five vertices are on a ray line.

So we're now at your task B where you are gonna be performing enlargement.

So for question one, you need to complete the transformation as described.

So read it carefully, make sure you're plotting your centre of enlargement accurately.

And remember, you can do this with counting squares, you can draw the ray lines first or you could use ray lines to check.

Press pause and then when you're ready for the next question, press play.

Okay, so here we've got question two.

It's a very similar question.

More enlargements of these objects, plot the points, read the scale factor carefully, and perform the enlargement.

When you're ready for question three, press play.

And so the last question here is question three.

There's three copies of the same diagram and that's because I would like you to find three different results.

You need to come up with a centre enlargement and a scale factor.

So that little cross is at the position 5, 7, and one of the vertices on the image needs to be in that position.

You can choose the vertex, you can choose the centre of enlargement and the scale factor.

There is an interactive geo profile that you could do if you are a device instead.

And you can have a go at trying to make sure you get the vertex on 5, 7.

So press pause whilst you have a go down doing that on paper or by using the interactive solution finder.

When you're ready for the answers, then press play.

So question one, you needed to complete the transformation.

So on part A, I've drawn the ray lines on, I haven't done that on B, but you could do it to check that you're in the correct position.

Your image should be a rectangle with dimensions three by six because the scale factor was three and our object was one by two.

Its position, its bottom left vertex should be 3, 3.

On part B, you needed to enlarge by a scale factor of two.

The centre enlargement was the coordinates 2, 0.

So check that you did plot that at 2, 0 rather than 0, 2 'cause otherwise your image will be in the wrong position and your image should be two by six because the object was one by three and the bottom right coordinate is 6, 2.

It may be that you need to pause the video so that you can check vertex by vertex that you've got that correct.

Onto question two, again, we were enlarging, but this time our centre of enlargements were not outside of the object.

So the centre enlargement on part A was actually inside of the object.

And so our object and our image are overlapping.

And on part B, the centre of enlargement was on a vertex.

So once again, they are overlapping and they share a vertex.

So you may wish to pause the video and again, check the vertices using the pairs of coordinates to see if you've got them in the correct position.

Question three was a little bit more open-ended.

There were multiple correct answers.

So here are some examples.

So you could enlarge by a scale factor of two from the point -1, 1, and then the top left vertex was in the position of 5, 7.

Another example would be to enlarge by a scale factor of three from the point 2, -2 and that would be the bottom right vertex that is on 5, 7.

And another one would be an enlargement by scale factor of three from the point half, 2.

5.

So there is the interactive solution finder where you could try out your answer and see if the vertex on the image is in that position of 5, 7.

So to summarise today's lesson on checking and securing understanding of enlargement with positive integer scale factors.

So to enlarge an object, you need a scale factor and a centre of enlargement.

All lengths are multiplied by that scale factor, both on the object and the distance from the centre of enlargement.

When we look to describe an enlargement, you need to state that the object has been enlarged.

You need to give the centre of enlargement and also the scale factor.

Ray lines can be used to locate the centre of enlargement when you are looking to describe an enlargement, but also to check the position of the image if you have performed the enlargement.

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