Lesson video

Hello, everyone.

I am Mr. Gratton and thank you for joining me for this lesson on transformations.

Today, we will be looking at how to carry out transformations on a shape either on paper or using technology.

Pause here to look at some of the definitions that will come in handy today.

First off, let's have a look at the transformations of translation, reflection, rotation, and enlargement altogether.

Jun says, after you transform an object such as this reflection, you can't transform it again because the image, the end result, has already been drawn.

However, Laura disagrees.

Laura claims that you can have multiple different images for the same single object, such as this rotation of 180 degrees from that original object.

Laura is correct.

It is possible to apply different transformations, translation, reflection, rotation, and enlargement to an object to get different images.

You can do this as many times as needed.

But to avoid a space getting confusing with images, it is helpful to label each image differently in order to identify which image is the result of a certain transformation.

For example, an enlargement giving this image that we will call image P or a translation giving a second image Q.

If a vertex on the original object is labelled A, then the corresponding vertex on its image can be labelled with an A dash or more accurately called A prime seen here.

However, if there is more than one image from that single original object, then A prime cannot be used again for a different image.

Instead, we can also label corresponding points as for example, A1, like this.

Okay, for this first check, an object has been reflected in the line, x = 8.

On the object is a vertex marked question mark.

Pause here to identify the corresponding vertex on the image.

After the reflection, the vertex corresponding to the vertex marked question mark, is D prime.

For this next check, the original object is translated by the vector negative 2, 9 to create a second different image called image two.

You'll need to draw or visualise the location of image two yourself.

Pause here to write down the coordinates of vertex D1, the corresponding vertex to D on image two.

And here is the location of image two image where the vertex D1 has coordinates of 8, 11.

Rather than applying multiple different transformations all to the same original object, we can also apply transformations to already drawn images.

For example, let's rotate this object by centre of 8, 12 to get this image A.

We can then take image A and then translate that to get image B.

Notice that for an image made from another image, we can use double prime notation such as the vertex at 13, 7 being X double prime.

For this check, reflect object A to create image B and then reflect that image to create image C.

Pause here to write down the coordinates of the vertex T double prime.

A is reflected to make this image B and then image B is reflected again to make image C.

Therefore, the vertex T double prime is at the coordinates 9, 5.

For this check, translate object A to create image B then enlarge image B to create image C.

Pause here to write down the coordinates of the vertex K double prime.

Here is image B and then here is the enlarged image C.

Therefore, K double prime has coordinates of 11, 6.

Great work so far, everybody.

Onto some independent practise.

For question one, pause here to perform four transformations, a reflection, a rotation, a translation, and an enlargement all on this kite, the object.

And for question two, pause here to perform multiple transformations on this object and then it's subsequent images.

Great stuff.

Well done on performing all of these transformations.

Pause here for question one to compare the locations of the images on screen to your own.

And pause here again to compare the images for question two to yours.

Brilliant work, everyone, on all of the transformations that you've created by hand.

But now let's have a look at how can use technology to help us visualise and perform a range of different transformations.

The software that we will be using is called Desmos.

This dynamic software allows us to create, move, and manipulate shapes including through transformation.

Search for Desmos geometry or click on this link to go to the correct page.

In order to draw any polygon, we need to follow these steps.

Let's have a look at them in action.

To create a polygon, go up to this button, the polygon tool, click on it.

Each click on the workspace will create one vertex of the polygon that you want to create like so.

To complete your polygon, in my case, a triangle, you go back to the first vertex and click again.

When you move your cursor off of the polygon, you can see that the shape has been completed.

Pause here to construct a pentagon on this workspace.

Next up, let's have a look at how we can draw a line, maybe for a line of reflection.

Let's have a look at this in practise.

The line tool located here defaults to drawing a line segment, like this.

Click once for one endpoint and then a second time for the other endpoint.

Clicking on this, the dropdown arrow to the right of the line tool gives you multiple different options, including a line, array, and more.

Now that we know how to create both a polygon and a line of reflection, let's reflect that polygon.

Let's see this in action.

First of all, go to the select tool at the top, the leftmost button, and then click on the object that you want to reflect.

Notice how this transform option appears.

Click on it then select Reflect.

Notice the instruction near the top, choose a line of reflection.

Select the line, line segment or ray that you've drawn, and when you hover over that line, a grey shadow appears behind it.

Click on the line to finish your reflection.

You can adjust the line or line segment by either moving the points to create different reflections or you can move the reflection by moving the line itself.

This will either give you different locations or different orientations to your final reflection.

But what if we needed to reflect over an edge, a side of an object, rather than a separate line of reflection? Let's have a look by first clearing our screen.

You can clear your screen in multiple different ways.

Here's a few.

You can either select on something and then on your keyboard, click Delete or press the bin button like so, or you can go to the gear icon on the far left, click on it and then click on Delete All to completely clear your workspace.

If you want to reflect over an edge or side of an object, we follow a similar process to creating a line of reflection.

First of all, select the select tool, click on the object, transform then reflect, and then hover over the side or edge of the shape that you want to reflect until that shadow appears and then click.

Notice how your image now shares a side with the original object.

Okay, quick check on all that we've seen.

Pause here to identify which of these buttons opens the menu for the line tool.

D is the correct option.

To choose a segment or array, you need to click the dropdown icon to the right of the main icon.

And next, pause here to identify which button allows you to select a polygon that you have already drawn.

To select anything on your workspace, you select this, the select tool.

Onto the next transformation, a rotation.

Let's have a look at it in action.

Imagine, I already have a polygon that I want to rotate.

To rotate it, we first of all select the select tool, then click on the polygon, then Transform, and then click Rotate this time.

Notice how it will prompt you to choose a centre of rotation.

I will choose my centre to be here, so I click on the workspace.

If the centre is a vertex of the polygon, then you just click on the vertex that is already there.

After defining your centre, a dialogue box will appear asking you to type in the degrees of your rotation.

I will type in the degrees of my rotation, 85 degrees, and then either click Enter or Go to complete my rotation.

Although notice how Desmos doesn't ask you whether you wanted to go clockwise or anticlockwise.

Desmos defaults to always doing an anticlockwise rotation.

How do we do a clockwise one? Well, there are two ways.

To perform a clockwise rotation, you could find the conjugate of your angle.

In other words, 360 degrees, take away the angle given.

This will always ensure you that the end result of this calculation is going to be clockwise.

Another way of doing this is by entering the negative of the angle that you want to rotate in a clockwise direction.

So imagine I wanted to rotate this object, 90 degrees, clockwise.

I can either type in 360 takeaway 90, which is 270 degrees, or I could type in negative 90 degrees.

We can create a variable angle rotation one where the orientation of the image changes with this simple dragging of a slider with one easy step.

We go through the normal process to rotate our object as usual.

Choose our centre, but rather than typing in a numerical angle in the dialogue box, we type in a letter such as a, and then click Enter or Go.

This creates a slider on the far left, which allows you to modify the exact value of our angle of rotation.

Notice how as the angle increases, as we drag our slider to the right, the image rotates in an anticlockwise direction.

Right, next check, true or false.

The object will rotate 60 degrees clockwise.

Pause now to choose an answer and justification.

This is false as the 60 degrees on Desmos always defaults to a 60 degree rotation in the anticlockwise direction.

But Jun wants a 60 degree clockwise rotation.

Pause here to consider what angle to type in and there are two possible answers.

Jun should type in either the result of 360 takeaway 60 or 300 or simply negative 60 degrees.

And next, pause here to describe what you would've typed in to make this rotation.

This is a 90 degree clockwise rotation, so for Desmos, that means a 270 degree anticlockwise rotation or a negative 90 degree rotation.

Okay, for our next transformation translation, a vector can be used to describe the movement from an object to its image.

For example, the vector 2 negative 5 represents the movement of two units to the right and then five units down.

Desmos represents vectors as a single arrow that considers both the horizontal and vertical parts of the column vector.

Let's see how we can draw a vector in Desmos.

As usual, click on the select tool, click on the object that you want to translate, transform, and then click Translate this time.

Click on any one of the vertices of your polygon and you'll see that a vector arrow will appear.

Click on any point on the workspace to plot the location of the corresponding vertex of the image.

When you click, the rest of the shape will be drawn.

Notice how in the image, only one single vertex, the corresponding vertex to the one that you used in the original object is shown with a point that is highlighted.

Right, let's see how much you can remember so far.

Pause here to match the symbol to its purpose.

Here are the answers.

In the UK, enlargement is the name of the transformation that causes a change in size, either larger or smaller.

However, in the USA, the word used is dilation, not enlargement.

In Desmos, the enlargement transformation is called dilate.

Right, let's apply these instructions to enlarge an object.

Let's have a look.

To enlarge our object, click on the select tool, click on the object, click Transform, and this time select Dilate.

Similar to rotation, select a centre of enlargement, which I will choose to be here.

After clicking, it will prompt you to type in a scale factor.

The scale factor can be any number, so I will type 2.

44, then click Enter or Go to activate your enlargement.

If you can't see your enlargement, then move the centre closer to the object and the enlarged image will also get closer to the object.

Similar to rotation, we can use a slider to vary parts of our transformation, this time with enlargement.

Let's see how.

After choosing the dilate transformation, we then choose a centre.

Rather than typing in a numerical scale factor, let's type in a letter for a variable scale factor.

In this case, I'll type in B.

Then either click, Enter or Go.

This, again, will create a slider.

As we drag the slider from left to right, our enlargement will get even bigger because the scale factor will increase.

Conversely, if we move the slider from right to left, the scale factor will decrease, meaning the size of our image will also decrease.

I wonder what happens when the scale factor is one and I wonder what happens when the scale factor becomes less than one.

Now that we've looked at four different transformations, pause here to match the button to the transformation that it creates.

Okay, here are the answers.

A is a translation, B is a rotation, C is an enlargement or dilation, and D is a reflection.

And a quick note, we can transform objects on a Cartesian coordinate grid.

This is especially useful if we're given the coordinates of the vertices of the object rather than the object itself.

To get up a Cartesian coordinate grid and axes, click on the spanner icon and then turn on both grid and axes.

Right, time for you to practise Desmos on your own.

Pause here to click on the file link for question one.

Create an image of the given object for all four transformations and try to match this picture a closely as possible.

And pause here to load up question two, follow the instructions on screen and apply them to the object on that Desmos file.

And finally, pause here to open up question three, follow the instructions on screen and apply them to the new object on this different Desmos file.

Great work Exploring on Desmos.

Pause here to open up the answers file for question one.

Your transformations should closely match these in the file and the picture on screen.

Okay, for question 2A, the vertices on the image should have been 13, 6, 17, 6 and 17, 13.

For part B, the single vertex that has coordinates that are completely positive is at 2, 3.

For question two C, the vector is negative 8, 2.

Pause here to open up the answers file for question two.

And finally, question three.

Part A, the vector is negative 2, 9.

A massive well done if you identified any of the scale factors, 0.

6, 1, 3 and especially that obscure one that is around 0.

43.

Now that we've practised all of these transformations, let's see if we can solve transformation problems either on paper or using Desmos.

Transformation instructions are not tied down to any one object.

In fact, these instructions can exist before any object is even drawn.

It is then possible to plot these vertices on a polygon and then apply transformation instructions to the polygon that you just created.

For example, a triangle has these vertices and is then plot.

What will the coordinates of the vertices of its image B after transformation A is applied and then transformation B is applied? Let's have a look.

Starting with the rotation and then the translation, its final image has coordinates of 9, 13, 12, 13, and 10, 11.

For this check, an object has vertices 0, 0, 0, 2 and 2, 2.

Pause here to apply the transformations, A, then B, and then A a second time to this object and write down the coordinates of its final image.

Here is the object.

Here is the image after transformation A, then B, and then A.

Again, giving us this, final image with vertices of 0, 6, 0, 8, and negative 2, 8.

Here are two more transformations.

Image one is created by doing transformation A and then B.

But what if we applied these two transformations in the opposite order? Transformation B and then A applied to the object.

Will the image end up in the same location as image one or in a different final location? Let's have a look.

Transformation B is first a translation.

But if I wanted to rotate this middle image onto image one, the centre of that rotation would have to be at 8, 12.

However, the instructions dictated that the centre is at 5, 7 instead.

So actually, the final image would be far away from image one.

This means that the order of transformations matters.

It is possible for the order to change and then the location of the final image change as well.

Both image one and two were created by applying two different transformations, but each one applied those transformations in a different order.

Rather than two transformations, is there one single transformation that would map the object onto these two images? Well, notice how orientations of image one and two are both different to the object, so a rotation would work, but each would need a centre that is different on image one versus image two.

To create image one, we would have a rotation of 180 degrees with a centre of 6.

5, 9.

5.

However, for image two, we would have also a rotation of 180 degrees because the orientation of both images are the same.

However, the centre this time would be 3.

5, 4.

5.

Furthermore, is it also possible to map image one onto two? Well, yes.

Notice how both image one and two have exactly the same size and the same orientation that we established earlier, but they are in different positions.

This means we can map image one onto image two using a translation.

Therefore, the translation by vector negative 6, negative 10 would map image one onto image two.

For this check, this is the location of image one and this is the location of image two.

Pause here to identify the vector that translates image two onto image one.

The vector is negative 4, 0.

Brilliant.

Onto the final practise task.

Grab some squared paper or open up a new Desmos file with both the grid and axes turned on.

For question one, pause here to plot the object a square and then apply these transformations to this object.

Pause here for parts A to D.

And for question two, do something similar, but this time, for an object that is a rectangle.

Pause here to answer parts A to E.

Brilliant.

Onto the answers.

For part A, the four vertices of the image of the square are 13, 4, 15, 4, 15, 6, and 13, 6.

Whilst for part B, the vertices are 5, 4, 7, 4, 7, 6, and 5, 6.

For part C, notice how applying transformation A and then B meant that the image overlaid the original object completely.

Furthermore, image one is a translation of image two by the vector negative 4, 0.

And for part D, a very well done if you're able to fully delve into the properties of these transformations.

Image one will always be a translation of image two.

However, unless the shape has a vertical line of symmetry, image two will not fully overlay the original objects.

For question two part A, the vertices of the image are 3, 2, 5, 2, 5, 6 and 3, 6.

And for part B, 3, 2, 5, 2, 5, 6 and 3, 6 as well.

For part C, well, you might have noticed that image one and image two overlay each other completely.

The vertices are identical.

For Part D, no matter what the shape is, even for a non quadrilateral, both image one and image two overlay each other completely.

And for part E, not always.

If the centre of rotation is on the x-axis, for example, 7, 0 or negative 3, 0, the image will overlap completely.

Otherwise, the image will never overlap completely if the centre is not on the x-axis.

Once again, some amazing work on performing and analysing all of these transformations in a lesson where we have looked at multiple different transformations being applied to one object and that we can apply a second transformation to an image made from a previous transformation.

We've also looked at Desmos as a way of transforming polygons created vertices.

And that the order of transformations being applied to an object and its subsequent images may change the location of that final image.

That is all from me, Mr. Gratton.

Thank you all so much for joining me in this lesson.

So take care and have an amazing rest of your day.