Lesson video

Hello, my name is Dr.

Rowlandson, and I'll be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the unit of Transformations.

This lesson is called Investigating Transformations with Desmos, and the aim of today's lesson is to learn how to investigate transformations using dynamic software, which in the case of today's lesson will be Desmos.

Here is a reminder of some keywords that you may be familiar with and we'll be using in today's lesson.

This lesson contains four learning cycles, with each learning cycle focusing on one of the four transformations and how to do it in Desmos, and to begin with, we'll be investigating translation with Desmos.

Dynamic geometry software in general can be used to quickly draw shapes and transform them.

It is called dynamic because it means that items can be moved and manipulated after they've been created.

This is in contrast to when we do transformations on paper with pencil and ruler, 'cause when we perform a transformation, after we've drawn the image, there's not a lot we can do to change it or manipulate it in any kinda way without rubbing things out and starting again.

So dynamic software allows us to move things around and manipulate them after they've been created.

This allows us to explore transformations by examining what happens when we change aspects of a diagram after a transformation has already taken place.

During this lesson, I'll demonstrate how to do lots of different things on Desmos, and for each demonstration, I'll first show you doing it on Desmos, and then I'll repeat the explanation with step-by-step instructions for you to have a go.

To get started, open up a web browser and go to desmos.

com.

That should bring up a page that looks like this.

You then need to click on the Open Geometry button.

You should now see a page that looks a bit like this, a big blank space here, which we call the workspace, where we can draw polygons and do transformations and other things and also a toolbar up here, which contains all the tools we'll use in today's lesson.

So to get started, open up a web browser and go to desmos.

com.

Then find and press the Open Geometry button, and you'll get a space that looks like this where you've got a toolbar where all your tools are and a workspace for drawing your polygons and doing your transformations.

Let's now draw a polygon on our workspace.

Under the toolbar, click on the polygon drawing tool.

Click on some points in a workspace to insert some vertices, and then when you're done, click on the first vertex again to complete the polygon.

Let's see that again with step-by-step instructions.

Click on the polygon tool.

Click on the workspace to insert some vertices, and then click on the first vertex again to complete the polygon.

With translation, a vector can be used to describe the movement in that translation.

For example, the vector 2 minus 5 represents a movement of 2 units to the right and 5 units down, and in the diagram on the screen, you can see that's two arrows, one arrow going 2 units to the right and another arrow going 5 units down.

Desmos will represent vectors as a single arrow going from the start to the finish.

So the arrow goes 2 units to the right and 5 units down, but it's shown as a single arrow.

Let's now insert a vector into our workspace.

Under the toolbars, we can see there's this button here which has a line between two points.

It's a line segment.

Next to it, click on the down arrow to bring up some options.

These are all various line drawing tools, and we want to draw a vector.

Click on a point for where you want the vector to start and a point for where you want the vector to end, and now let's do that again step by step.

Click on the down arrow on the line segment button to open up the option menu.

Choose the vector tool, and then click on two points on the workspace to insert your vector.

Let's now translate this polygon according to the vector on this workspace.

Firstly, you can see that my cursor still has the spot on it from when I last drew my vector.

If I want to escape the last two I used, I need to first click on the select button.

Now let's click on somewhere on the polygon, and we can see the toolbar has changed up here, and it's got this transform button here.

Click on the down arrow to bring up some options, and we wanna choose translate.

It says choose a point or a vector, so let's click on this vector here, and that will tell Desmos we want to translate this polygon according to the movement described in this vector here.

Now let's see that again with step-by-step instructions.

To translate an object, click on the select tool to begin with.

Click on the polygon, and then click the transform button to open up an options menu.

Click on the little arrow if it doesn't open first time.

Choose the translate tool, and then click on the vector.

Now we've done that, we can describe your items in the workspace using our mathematical terminology.

So the vector shows the movement between the object and the image.

In this case, it's moved right and then down a bit.

The new polygon that is created by a transformation is called the image, and the polygon that we selected to transform is called the object.

Let's check what we've learned.

Which of these buttons allows you to draw a polygon? Is it a, b, c or d? Pause, have a go, and press play when you're ready for the answer.

The answer is d.

That tool is the polygon drawing tool.

Which button opens up the option menu containing the vector tool? Is it a, b, c or d? Pause the video, make a decision, and press play when you're ready for the answer.

The answer is b.

That is the line segment button, and when you click on it, it allows you to select the vector tool, and true or false? The transform button is always displayed in the toolbar.

Decide is that true or false and also justify your answer.

We have two justifications below.

A is you need to click on the object before you can see the transform button, and justification b is you need to click on the transform button before you can click on the object.

Pause, decide now, and press play when you're ready for the answer.

The answer is false because you need to click on the object before you can see the transform button.

Over to you now for Task A.

This task contains three questions all using what we've learned so far.

First, you need to open up Desmos, draw any polygon, and translate it to the right using a vector.

Once you've done that, here are some questions to think about.

Question 1 is click on your object and drag it around the workspace and see what happens to the image.

Write down what you notice happening to the image when you drag the object around the workspace, and write down why do you think it does this.

Question 2 is click on the end of your vector and drag it to make the vector longer.

Write down what you notice that happens to the image when you do that, and write down why does it do this.

Then question 3, click on the middle of your vector and drag it around the workspace.

What happens to the image when you do that, and why does it do this as well? For each question, write a sentence or two.

Pause, have a go, and press play when you're ready for some answers.

Let's work through Task A together now.

To start with, we need to draw any polygon in the workspace, and we then need to translate this polygon using a vector that goes directly to the right.

Now we've done that, question 1 says, "Click on the object and drag it around the screen and see what you notice happening." Let's do that now.

The image always moves with the object.

Wherever the object is, the image is always positioned to the right of it because the vector that we use to translate moves the image to the right of the object at all times.

In question 2, we had to click and drag the point at the end of the vector to make the vector longer and see what we notice.

As we do that, the image gets further away from the object, and that's because we've increased the distance covered by the vector.

So the distance increases between the image and object.

In question 3, we click on the middle of the vector and move it around and notice it has no effect on the image.

That is because the vector still has the same length and direction, and here are written answers for questions 1 and 2, and here's a written answer for question 3.

Well done so far.

On to learning cycle two, which is investigating rotations with Desmos.

Let's look at how to rotate an object using Desmos.

To begin with, make sure the select tool is highlighted.

Click on the object.

Click transform and choose rotate.

It says choose a point or centre, so click somewhere on the workspace to insert a centre of rotation.

It then says construct or choose an angle.

This is where I'm gonna put the angle I want to rotate my object.

I'm gonna enter 90 degrees.

We can see an arrow now has appeared around the centre rotation, and the arrow is going anticlockwise.

That's because Desmos always rotates anticlockwise.

Worth bearing in mind.

Click go.

Let's go through that again with step-by-step instructions.

Using the select tool, click on the polygon, and then click the transform button.

Choose the rotate option.

Click a point on the workspace to plot a centre of rotation, and then type in an angle to rotate the object anticlockwise and press go.

Now, Desmos only rotates shapes anticlockwise, but not to worry because clockwise rotations can be replicated by subtracting the angle from 360 degrees.

For example, if I want to rotate the object 90 degrees clockwise, I could do that by subtracting 90 degrees from 360 to get 270 degrees.

A 90 degrees clockwise is the same as a 270 degrees anticlockwise rotation.

In the previous example, we're able to manipulate a lot of things about the object and the rotation after we completed the rotation, but one thing we couldn't change then was the angle of rotation.

We'll go ahead do this in a slightly different way now that allows us to create a variable angle of rotation, and we're gonna do it by inserting a slider.

So we'll start in the same way, but this time, where it says construct an angle, rather than putting in a constant number like 90 degrees, I'm gonna put in a variable, a letter.

I'm gonna press go.

It has now created this slider on the left here where it says a equals 90 in this case.

A is my angle of rotation, and I can adjust my angle of rotation after I've plotted it.

Let's go through that again step by step.

So start by carrying out the same steps to rotate the object as before, but when we get to the point where we need to enter an angle, rather than enter a number, enter a letter to represent a variable for the angle.

When we do this, a slider appears in the left panel, which we can move left and right to change the angle of rotation.

Let's check what we've learned there.

On the screen, you can see a snippet from Desmos from midway through doing a rotation.

True or false? The object will rotate 60 degrees anticlockwise when we press go.

Decide whether it's true or false, and then decide on a justification.

Your choices are, a, Desmos rotates objects clockwise, and b, Desmos rotates objects anticlockwise.

Pause the video, make your decisions, and then press play when you're ready for answers.

The answer is true because Desmos rotates objects anticlockwise.

Jun wants to rotate the object 60 degrees clockwise.

Desmos only rotates anticlockwise.

So what angle should Jun use to do that? Pause the video.

Write down what angle you think he should enter, and then press play when you're ready for the answer.

The answer is 300 degrees, which we can get from doing 360 subtract 60.

Over to you now for Task B.

To set up, open up Desmos.

Draw any polygon.

Whatever shape it is, it's up to you.

Rotate it using a slider, and set the slider to 270 degrees.

Here are now some questions to think about.

Question 1, drag the centre of rotation towards the image and notice what happens to the image.

Why do you think it does this? Question 2, is it possible for the centre of rotation to be inside the image? If so, how does it happen? And if you think it's not possible, why not? And question 3, move the spot in the slider left and right.

For what angles does the image perfectly overlay the object, and why does it do it for these particular angles? Pause the video.

Have a go.

Write a sentence or two for each question, and press play when you're ready for the answers.

Okay, let's go through Task B together now.

To set up, it said draw any polygon.

Then it said rotate it using a slider and set the slider to 270 degrees.

Now on to the questions.

Question 1 said drag the centre of rotation towards the image.

What happens to the image and why does it do this? Let's do that now.

Here's my centre of rotation.

If I move it up now, ah.

The image seems to move further away.

Why does it do this? Well, it's because the centre of rotation is moving further away from the initial object.

Therefore, the distance from the image to the centre of rotation also increases as well.

It's pretty hard to get the centre of rotation onto the image.

That's what question 2 says.

Is it possible for the centre of rotation to be inside the image? If so, how? If not, why not? If I keep trying to move the centre of rotation towards the image, I'm gonna be chasing it around, but I can make it work by moving the centre of rotation onto the object somewhere.

When the centre of rotation is inside the object, it's also inside the image as well, and then question 3 said move the slider to the left and the right.

For what angles does the image perfectly overlay the object, and why does it do it for these angles? Well, to begin with, let's just put my centre of rotation back out here.

Move the slider to the left and right.

I'm looking for when the image perfectly overlays the object.

There.

So it perfectly overlays the object for 0 degrees because it's not turned at all here.

Let's try the other way.

And there, the image perfectly overlays the object for 360 degrees because it's done a full turn.

In fact, if this slider went further, if I drag it to 720 degrees, you can see that every time I do a full turn, it'll overlay perfectly the initial object.

So for any angle that is a multiple of 360 degrees, it will perfectly overlay the object.

Here are the answers to questions 1 and 2 again in a written form, and here are some things you could write about question 3.

Well done.

You're doing great with this.

Let's now move on to learning cycle three, which is investigating reflections with Desmos.

Let's now reflect this polygon over one of its edges.

To begin with, make sure the select tool is highlighted.

Click on the polygon.

Click transform and choose reflect.

It then says choose a line of reflection.

I could choose any of these lines, any of these edges to be the line of reflection.

Click on the one you want.

Let's go through that again step by step.

Start with using the select tool.

Click on the polygon and click transform.

Choose the reflect tool.

Click on one of the edges of the object to reflect the image over that edge, and there we go.

We've got our image.

It might be that you don't want to reflect the object over one of its edges and you want the line of reflection to be somewhere off the object.

In that case, we first need to insert a line somewhere in our workspace.

So go up to the line segment tool here.

Click on the arrow to bring up some options.

Choose line, and then click on two points in the workspace to insert a line.

Let's do that again.

So click on the down arrow on the line segment button to open up an options menu.

Choose line, and then click on two points in the workspace to draw a line through those two points.

To reflect an object over a line, first draw a line, and then as before, click on the object.

Click transform, reflect, but this time, click on the line you want to reflect over.

Let's review that again.

First, draw a line.

Then click on the object and select reflect, as we did before.

Then click on the line, and it's reflected the shape over the line.

We can then adjust the line by moving the points around and seeing what it does to the image.

Let's check what we've learned.

Which of these buttons opens up the option menu containing the line tool? Is it a, b, c or d? Pause the video, make a decision, and press play when you're ready for the answer.

The answer is d.

That is the line segment tool at the moment, but when you click on it, you can choose line tool instead.

True or false? You always need to draw a line before you reflect an object.

Decide whether it's true or false, and then choose one of the justifications.

Justification a, the line tells Desmos where to reflect the image over, and justification b, you can reflect over one of the edges of the object.

Pause the video, have a go, and press play when you're ready for an answer.

The answer is false because you can reflect over one of the edges of the object.

Over to you now for Task C.

For this, you need to set up by opening up Desmos, and draw a polygon where every edge has a different length, for example, a scalene triangle, and then draw a vertical line and reflect the polygon over that line.

Once you've done that, here are some questions to think about.

Question 1, drag the line of symmetry towards the image.

What happens to the image when you do that, and why does it do it? Question 2, is it possible for the line of symmetry to be inside the image? If you think it is possible, how? If you think it's not possible, why not? And question 3, is it possible for the image to perfectly overlay the object? If you think it is, how do you do it, and if you think it's not, why not? Pause the video, have a go, write a sentence or two for each question, and press play when you're ready to go through it together.

Okay, let's go through Task C together now.

To set up, it said draw a polygon where every edge is a different length, for example, a scalene triangle.

It said draw a vertical line and reflect the polygon over the line.

Then question 1 said drag the line of symmetry towards the image.

What happens to the image, and why does it do this? As I drag the line of symmetry towards the image, the image gets further away.

The reason why it does this is because the line of symmetry is always equidistant from the object to the image, which means the distance, for example, from this vertex to the line of symmetry is the same as the distance from this vertex to the line of symmetry as well.

So as I move the line of symmetry further away from the object, the image gets further away as well.

Question 2 said, is it possible for the line of symmetry to be inside the image? If so, how? If not, why not? Well, moving the line of symmetry towards the image isn't gonna work 'cause it keeps increasing the distance from the object to the line, but if I move the line of symmetry towards the object, we can see that the image is getting closer until I get to the point where the line of symmetry is inside the object.

It is also now inside the image as well.

So the answer to that one is yes.

When the line of symmetry is inside the object, it is also inside the image.

Question 3 said, is it possible for the image to perfectly overlay the object? If so, how? If not, why not? In this case, no, it's not possible for that to happen because when the object is reflected, it doesn't retain the same sense of the object.

The image is pointing the opposite direction to what the object is.

So it never quite perfectly overlays the object.

So here are some of the things you might write about questions 1 and 2, and here is what you might write about question 3.

On to the last learning cycle of today's lesson, which is investigating enlargements with Desmos.

Before we enlarge any shapes with Desmos, we need to note something about terminology differences in different countries.

In the UK, the word enlargement is used for a transformation that causes a change in size, and that can be any change in size, stretch or a shrink.

However, in the USA, the word dilation is used for a transformation that causes a change in size instead.

Desmos was created in the USA, so it uses the word dilate to mean the transformation that we call enlargement.

Here is how to enlarge an object.

We're gonna start in the same way by ensuring the select tool is highlighted, clicking on the object, clicking on transform, and this time, we're gonna choose the dilate tool.

It says choose a point or centre.

Click somewhere on the workspace to plot my centre of enlargement.

I'm gonna plot it here.

It then says scale factor.

Type in a number for your scale factor, and then press go.

So I've got the number 2.

Let's change it to a scale factor of 3.

Let's do that again with step-by-step instructions.

Using the select tool, click on the polygon, and then click transform.

Choose the dilate button.

Click a point somewhere on the workspace to plot a centre of enlargement.

Type in a scale factor and press go.

We can make the scale factor in an enlargement dynamic by inserting a variable for the scale factor, which creates a slider, and that means we can adjust the scale factor after we've plotted our image.

To do that, we'll go start in the same way.

Let's enlarge the shape.

And this time, where it says scale factor, then we'll insert a letter for a variable and press go.

We can now see it's created a slider here, which initially sets a as 2.

So our scale factor is 2 here, but I can drag my slider to the left and right to adjust a scale factor.

Let's go through that again step by step.

Carry out the same steps to enlarge an object as we did before, but when we get to the point where we need to enter a scale factor, rather than typing in a number, type in a letter to represent a variable scale factor.

That will create a slider in the left panel, which we can drag left and right to change the scale factor.

Let's check what we've learned.

What word can be found on this transformation button? Is it a, dilate, b, enlarge, or c, move? Pause the video, make a choice, and press play when you're ready for the answer.

The answer is a, dilate.

Over to you now for Task D.

You'll need to set up by open up Desmos.

Draw any polygon and enlarge it with a slider scale factor.

Once you've done that, here are some questions to think about.

In question 1, drag the slider left and right and look for, a, when is the image larger than the object, b, when is the image the same size as the object, c, when is the image smaller than the object, and then d, what happens to the image when the scale factor is 0, and why does this happen? And then with question 2, select the line tool.

Click on a vertex on the object and its corresponding vertex on the image to draw a line that traverses through both of those vertices, and repeat this for all pairs of corresponding vertices.

Then look.

Where do the lines intersect each other? Pause the video, have a go at all these, and then press play when you're ready to go through it together.

Okay, let's go through Task D together now.

To set up, it said draw any polygon and enlarge it with a slider scale factor.

And then question 1 said drag the slider to the left and right and look out for the following: when is the image larger than the object, when is the image the same size of the object, when is the image smaller than the object, and what happens to the image when the scale factor is 0, and why does this happen? Let's do that now.

Let's start by dragging the slider to the left and right.

We can see that sometimes the image is bigger than the object, sometimes it's smaller than the object, and sometimes it's the same size and perfectly overlays the object.

When does it do each of these things? The image is larger when the scale factor is more than 1 because we're multiplying all the lengths and distances by a number that is greater than 1 to give a greater result, just like when you multiply a number by something that's more than 1, you get a bigger answer.

When is the image the same size of the object? That's when the scale factor is 1 because we're multiplying all the lengths and distances by 1, which gives the same result, just like when you multiply a number by 1, you get the same result, and when is it smaller than the object? That's when the scale factor is less than 1 because multiplying by a number between 0 and 1 gives a smaller result, and then what happens when the scale factor is 0? Well, when we multiply by 0, we always get 0.

So let's see what happens here.

When the scale factor is 0, the image seems to disappear.

That's because it has no size.

All those lengths have been multiplied by 0 to give a length of 0, so the image has no size.

Question 2 said select the line tool.

Click on a vertex on the object and its corresponding vertex on the image to draw a line that traverses through both vertices, and then it says repeat for all pairs of corresponding vertices.

And then it asks, where do the lines intersect each other? All of these lines, I've got a lot here, so you can definitely see this for sure, all of these lines intersect around at the centre of enlargement, and now here are the answers again.

Fantastic work today.

Great job with that.

Here's a summary of what we've learned in today's lesson.

Firstly, dynamic geometry software can be used to quickly draw shapes and transform them.

Dynamic software also allows us to manipulate aspects of the shapes after we've done the transformations, and sliders can be used to make values such as angles and scale factors dynamic.