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 learning outcome today is to be able to describe an enlargement.

There are some key words that I'll be using during the lesson, which you will be familiar with from previous studies, but you may wish to pause the video and read those definitions again to make sure you're confident before we make a start with the lesson.

So our lesson is describing a negative enlargement and we're gonna split this into two learning cycles.

So the first learning cycle is about identifying the type of scale factor that has been used, and then we're gonna go onto the second learning cycle where we'll describe the enlargement fully.

So let's make a start at thinking about scale factors by themselves.

So when a shape is enlarged, the object and image are similar to each other.

The angles are unchanged, but the length change dependent on the scale factor.

So it is true that if you have a pair of similar shapes, then it might have been an enlargement.

So if we focus on this first enlargement on the diagram where A is mapped onto A prime, so that's the image of A, then what do we notice? Well, we notice that the image is larger than the object and that is because the absolute value of the scale factor will be greater than one.

So regardless of whether the scale factor is positive or negative, if you multiply an edge length, let's call it one, by a value that is greater than one, then it will increase.

And that's why this image is bigger than the object.

If we now look at the B that maps onto B prime, the image of B, here in the box, what do we notice with this one? Well, this one, the image is actually smaller than the object.

And so, that tells us that the absolute value of the scale factor will need to be less than one in order for it to get smaller.

Because if you multiply a value, let's say one, by a number that is less than one, for example 0.

8, then the product will be 0.

8.

It is less than the original amount.

And lastly, if we look at this enlargement where C has mapped onto C prime, well, what do we notice? Well, we notice here that the object and the image are actually congruent.

They're exactly the same size.

There isn't one bigger than the other, they are the same.

And so, this would mean that the absolute value of the scale factor is equal to one.

So these are the three cases that we will have, we'll either have the image being bigger than the object, and therefore the absolute value is greater than one, or we'll have the image is smaller than the object and that means the absolute value of the scale factor is less than one or the object and image will be congruent, which means that the absolute value is equal to one.

So we'll check on that for you.

After an enlargement, the image is bigger than the object.

That means that the absolute value of the scale factor is less than one, equal to one or greater than one? So pause the video, it may be that you want to rewind the video and rewatch that last explanation, but when you are ready to check your decision, press play.

So that would be greater than one.

So if the image is bigger than the object, then we know that the absolute value of the scale factor will be greater than one.

So the position of the image is dependent on the centre of enlargement.

So when the centre of enlargement moves, the image will also move.

However, the scale factor does play a part as well.

And so we're gonna have a look at that bit now.

So when the centre of enlargement is not on the object and we have our object, so here the object is our scale factor one, if you like, that's our unit, that's the one we're comparing to, if the scale factor is greater than one, is positive and greater than one, then we're going to have an image that is bigger than the object, it would be located here.

If the scale factor was two, then it would be here.

It's still bigger than the object and that's because the absolute value is greater than one.

If it was three halves or 1.

5, still bigger and it would be located here.

Five quarters, the absolute value is greater than one, so the object is still smaller than the image.

The image is still bigger than the object.

If the scale factor is one, then we have a congruent shape.

The object and image are identical and also they would be in exactly the same position.

You wouldn't be able to see an object and an image.

Now if we have a scale factor of three quarters, well, this is where our image starts to become smaller because it is an absolute value that is less than one.

A half would be smaller, three eighths would be smaller still, one quarter would be even smaller.

Then we might have some negative scale factors.

So negative a quarter is smaller than the object because the absolute value of negative a quarter is one quarter and one quarter is less than one.

And therefore our object and our image have this relationship, that the image is smaller and this will happen for negative half.

And negative one is when we start to see a congruent shape again because the absolute value is equal to one, and therefore the object and the image are congruent.

Now we move into scale factors which will make a bigger image than the object.

So three halves, the absolute value of that is greater than one, and therefore the image is bigger.

Same for negative two and the same for negative 12 fifths.

So how can we spot that the scale factor is negative? We've got an idea of how we know whether the scale factor will be greater than one or less than one or equal to one or the absolute value of that scale factor.

And that's to do with comparing the size of the object and the image.

But how do we know that the scale factor will also be negative? And the way we'll spot that is if the orientation has changed.

So if the orientation has changed and the object and image are rotations of each other by 180 degrees, then the scale factor will be negative.

So which of these are enlargements with a negative scale factor? So pause the video, make a decision, and when you're ready to check, press play.

So that would be C.

On A, we've got a pair of similar shapes, but the orientation hasn't changed.

On B, we've got a pair of similar shapes, but the orientation hasn't changed.

Whereas on C, we've got a pair of congruent shapes, which means that the absolute value of the scale factor is equal to one, but they are a rotation of 180 degrees of each other, and therefore the scale factor is negative.

So Andeep and Jacob are discussing the transformation that has taken place on this parallelogram.

So Andeep says that this is a translation by two negative six.

So a translation is another type of transformation and that's where the object is moved horizontally and or vertically.

And he has given the column vector for that movement, which is two minus six, which means two to the right and six down.

Jacob says "No, this is not a translation, this is an enlargement by a scale factor of negative one." So who is correct? So just take a moment to look at that diagram.

Think about their two statements and who is correct.

So Jacob is correct, and the reason we know that Jacob is correct is because the orientation has changed.

We know the orientation has changed because of the labels of the vertices.

On the object, it was A, B, C, D.

And if we start on the image at the bottom left vertex, it's C prime, then D prime, then A prime, then B prime.

So that order is the same, which means it's a rotation because they're not in the same relative position but they are in the same order.

So this is a scale factor of negative one.

We know it's one because they are congruent, the size hasn't changed, but because there is a rotation of 180 degrees, then it would be a negative.

So for you three statements there, two blanks to fill.

So pause the video, read the sentence, work out the missing word, and then press play to check your answers.

So the first missing word would be positive.

So if we read this, "An enlargement has taken place, if the orientation of the object and the image are the same, then the scale factor is positive." So if the orientation has not changed after an enlargement, then we know that the scale factor would be positive.

The second statement says, "If the image is larger than the object and the orientation is different, then the scale factor is less than negative one." So we don't need to worry about absolute because we know that it will be negative.

So our scale factor will be less than negative one.

So on to the first task of the lesson where you on question one need to match the given scale factors to the images, and then there are two blanks which you need to complete the missing scale factors.

They are drawn to scale, so you can use that to support you with matching up the images of the darker shaded ones and the object is labelled as A.

So pause the video whilst you do that matching exercise.

And when you press play, we'll go to question two.

So question two, the object B has been enlarged.

I would like you in the two-way table to draw an example of the image of B and label it B prime for each region within the two-way table.

So our column headers are about the absolute scale factor and our row headers are to do with just the scale factor, not the absolute scale factor.

So pause the video and when you press play, we're gonna go through our answers to task A.

So question one, you needed to match the scale factors to the correct image.

And the ones that had not got a scale factor, you needed to write down what it would be.

So they were drawn to scale and that was gonna help you know the missing scale factors.

So anything that had an orientation change, was a rotation of 180 degrees, should have a negative scale factor, and any of the images that were the same orientation as the object should be a positive scale factor.

We know that the largest one will have the highest absolute value.

So that's why three would be put on the very largest image.

And then the two and the negative two, they are congruent to each other.

If you were to measure them or use a piece of tracing paper and lay it on top, they were congruent to each other.

The only difference was the orientation.

So that's why we know that the absolute value would be the same, but one was negative and one was positive.

We can use the same concept with the smallest two images.

And so, they would be the fractional ones, 0.

6 is three fifth, so that is fractional, it's less than one, so that is why it's smaller than the object and one's positive and one of them is negative because of the orientation.

And then the last one, negative one, that is congruent to the object, which means that the absolute value would be equal to one.

However, it's a rotation of 180 degrees, which makes it negative one.

On question two, you needed to draw an example of the image.

So they didn't have to be exactly the same size as what I have put in the two-way table except from the central column.

Because the central column was where absolute scale factors were equal to zero, which means that the image would be congruent to the object.

So it did have to be exactly the same size as my given object.

So on the first column, the absolute scale factor was less than one, which means that the image will be smaller than the object.

So it didn't matter how small you drew them, as long as it was smaller than the object, And the last column, it needed to be greater, they needed to be bigger.

And that's because the absolute scale factor was greater than one.

So you should have drawn a larger image on that column.

And then when we look at the rows, the scale factor being less than zero means that the scale factor is negative.

If it's negative, then the orientation has changed.

So it should be rotated by 180 degrees.

So if we think about the isosceles triangle pointing up on the object, then all of these are pointing down on the image, whereas the bottom row, they should all have the same orientation as the given object because the scale factor is positive.

So in our second learning cycle, we are now gonna pull that information about identifying scale factors and write a full description for a negative enlargement.

So we've got Izzy, Laura, and Alex having a bit of a discussion about how do we know an enlargement has taken place.

So Izzy says, "How can I recognise an enlargement?" So when you are faced with a question where a transformation has taken place, you have an object and you have an image, how do you identify it to be an enlargement rather than a reflection, a rotation or a translation? Well, Alex says, "Well, the object and image will be different sizes." So one way we're gonna spot an enlargement is that we have different sized objects to an image.

Laura's helpfully reminded them that that's only true if the scale factor is not negative one or one because as we know, they produce congruent images.

So on the most part, different size object and image is going to be the clear indicator that an enlargement has taken place.

But be mindful that sometimes an enlargement has taken place and the size hasn't changed and they would be because of the scale factors negative one or positive one.

So here we have a diagram with a transformation and Izzy has noticed and said, "This is an enlargement." So she's used their help to say, okay, so if they're different sizes, then this likelihood is this an enlargement.

And Alex says "Yes, and we know the scale factor is negative." So is Alex correct? I'd like you just to think about that for yourself.

Is Alex correct? So yes, and the reason we know the scale factor is negative is because the orientation.

They are rotated by 180 degrees of each other.

Izzy even remembered that we also need a centre of enlargement when we are describing.

So she's drawn ray lines to locate the centre.

Alex has said, "You've not quite done that right, because they need to be lines that connect the corresponding vertices." So if we had labelled the vertices of the quadrilateral, then we would look in at the image of that corresponding vertex.

So it should be more like this and that looks better because they are all passing through the same point, of which that point will be our centre of enlargement.

So Izzy said, "So object A has been enlarged from negative one, negative three." So that's the coordinates for that point.

Alex has reminded Izzy, yes, but with a description, we also need the scale factor.

So when we're describing an enlargement, we need to state that it's been enlarged, we need to give the centre of enlargement, but we also need that scale factor.

Izzy is looking at corresponding edges.

It's really important that you are comparing corresponding lengths that might be the edge of a shape, it might be the perpendicular height of a shape, but you need to make sure you are comparing corresponding lengths.

And she said that, "The corresponding edges are twice as long on the image." So that Alex says, "Well, that means the scale factor is negative two." So here we have already processed that the scale factor is negative because the change in orientation.

So now we can compare, how do you get from an object length of two to an image length of four? Well, you would multiply by two.

So that's why the scale factor is negative two.

And so, a description now has increased with the amount of information, object A has been enlarged by a scale factor of negative two from the point negative one, negative three.

And Alex says, "Exactly." So that is a full description of an enlargement that happens to have a negative scale factor.

So firstly, how do we know it's an enlargement? Well, we're looking for similar shapes and on the most part, the size of the object and the image will be different.

Then we need to locate the centre of enlargement and we can do line segments between corresponding vertices that will meet at the centre of enlargement.

And then lastly, we need the scale factor.

And so, to think about whether the image is bigger than the object, whether it's smaller than the object, whether it is the same size as the object, and also whether it's an orientation change or not.

So whether it's positive or whether it's negative.

So for you, can you complete this description? Press pause whilst you do that, press play to check your answers and then we'll continue with the lesson.

So the missing scale factor was negative 0.

5.

So again, we can say we know this is negative because the orientation has changed, they do not have the same orientation.

They are a rotation of 180 degrees of each other.

How do we know it's a half? Well, we know that it's smaller.

We know that the image is smaller than the object, which does tell us that the absolute value is less than one.

So we know we're sort of thinking about proper fractions or decimals less than one.

And so now we need to compare some corresponding lengths.

For ease, I would compare the vertical edge on both the object and the image.

So on the object, the vertical edge is four squares and on the image it is two squares.

So how do you get from four to two? Well, you multiply by a half.

So that scale factor is negative 0.

5 or negative half.

We can see where the centre of enlargement is because we've got those line segments and that's negative five, two.

We haven't got all of the line segments drawn on here.

There are four vertices, 'cause it's a quadrilateral, but there's only three line segments.

Two is all that you actually need, but a third one will check that nothing's gone slightly wrong.

So we're on to the last task of the lesson where you need to on question one, identify what is missing or incorrect in each of these descriptions.

So descriptions have been written for the enlargement in both A and for both B, but there's either something missing on the description or an incorrect element of the description.

So you need to identify that.

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

Question two, you need to complete the descriptions and match them to the relevant diagram.

So there are three diagrams and three descriptions, so you need to complete them and match them up.

Pause the video whilst you do those two things.

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

On to question three, there are four enlargements on here and you need to the full descriptions for each one.

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

So on question one, you need to either identify an incorrect element of the description or identify what was missing.

So on A, there was an incorrect element.

The scale factor had been written as positive half when it should have been negative half.

We know it's negative from looking at the fact that the orientation has changed between the object and the image.

The half part is correct, and again, we can just take a moment to think my image is smaller than my object, and therefore I know the absolute value will be less than one.

On B, they said that shape B has been enlarged by a scale factor of negative 1.

5.

We can check that.

Negative makes sense, there is an orientation change, they're rotated by 180 degrees of each other, 1.

5 while the image is bigger than the object.

So we do expect our absolute value to be greater than one and you can check it with corresponding lengths.

What was missing was the centre of enlargement.

And it's really important that you state the centre of enlargement because if the centre of enlargement was to be in a different position, then the image would also be in a different position.

The size of the image would not change, but its location would.

So you must state the centre of enlargement, and that would be four, three.

Question two, you need to complete the descriptions and match to the diagram.

So on the first description, that was describing the second enlargement on the diagrams and the missing part was the scale factor, so minus 0.

5.

On the second description, that was about the last diagram and it was missing the centre of enlargement, so that would be zero, two.

And the final description was missing the scale factor and the centre, and so that was a scale factor of negative two.

And the point to the centre was negative three, three.

Question three, you needed to write your full descriptions.

So for A, C and D, the centre of enlargement was given.

So for part A, shape A has been enlarged by the scale factor negative two from the point negative five, five.

So you needed to work out the scale factor.

Again, we know it was negative because the orientation has changed and the image is bigger.

So we know that the absolute value would be greater than one.

Compare corresponding edges, and you'd have been able to work out that was negative two.

For part B, you needed to locate the centre and that would be using the ray lines or the line segments.

So two line segments joining corresponding points is enough to locate that centre, which is at ten, three, but a third will help you check.

So you don't have to do all of them, but at least two and a third is a good practise just to make sure they're all passing through that same centre.

So your description would be shape B has been enlarged by the scale factor negative a third from the point ten, three.

Negative because the orientation has changed and we know that it's gonna be less than one because the image is smaller.

On to C and D, so shape C has been enlarged by the scale factor negative one from the point negative six, negative four.

The object and the image are congruent.

That tells us that the absolute value is going to be equal to one.

It was a negative one because there has been a rotation of 180 degrees.

For part D, shape D has been enlarged by the scale factor negative three from the point six, negative three.

So here the image is bigger, and therefore we know that the absolute value is gonna be greater than one.

It's negative, it's hard to tell that a rotation has taken place because of it being a rectangle, but from its location as well, we can see that that is negative.

So to summarise today's lesson, which was to looking at describing a negative enlargement, we need to make sure we state three things when describing any enlargement.

And that is state that the object has been enlarged, you must give the transformation that has taken place, give the scale factor and the centre of enlargement.

So those three things need to be present in your description.

If the scale factor is negative, then the image will be a rotation of 180 degrees of the object.

And so, that is a clear way of knowing that you're dealing with a negative scale factor is if you can tell that the orientation has changed and that would be a change of rotation of 180 degrees.

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