Lesson video

Hello, my name is Dr.

Rowlandson and I'm delighted that you'll be joining me in today's lesson.

Let's get started.

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

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

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

This lesson contains three learn cycles, with the first two learn cycles focusing on how to perform pairs of transformations with GeoGebra.

And the third learn cycle will be applying everything that we've learned throughout the lesson to learn how to identify transformations using GeoGebra.

But let's get started with learn how to perform rotations and enlargements with GeoGebra.

Dynamic geometry software can be used to quickly draw shapes and to quickly transform them, and it's called dynamic because 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, because when we do it on paper, once we perform the transformation, we can't really change things about the object or the image without rubbing out and starting again.

But on dynamic software such as GeoGebra, we can do that.

We can create our image and then manipulate it afterwards and see what we notice.

It also means that if we perform a transformation incorrectly to begin with, we don't have to start all over again.

We can just adjust it to make some corrections.

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

org and then click on the start calculator button to open up a new GeoGebra page.

This has opened up a graphing page, whereas today we want to use a geometry page.

So go to the graphing button, click on the arrow to open up some options and click on geometry.

We now have a big workspace for drawing shapes and doing transformations and some basic tools.

Currently the workspace has a blank background.

If we'd like to have a grid in the background, click on the cog and then click on show grid.

And we can choose a few options.

We can choose major and minor grid lines, which has all these little squares in between the big squares.

But today, let's go to major grid lines only.

To insert a polygon, find the polygon tool under basic tools.

Click on some points in the workspace to insert some vertices, and once you've insert all the vertices you want, click on the first vertex again to complete the polygon.

When you first open up a GeoGebra geometry page, it only has a few basic tools available.

If you want more tools on this, first click on more, and it brings up a lot more options.

If you click on more, again, it brings up even more options.

Let's go through that again, step-by-step.

Open up a web browser and go to geogebra.

org, find and press the start calculator button.

That opens up a graphing page.

So we need to find and press the graphing button and select from the options menu, geometry, to open up a geometry page.

If we want to change the background once we've done that, we need to first click on the cog in the top right hand corner of the screen and then click on show grid.

And from the dropdown menu, select major grid lines.

Then to draw a polygon, click on the polygon tool, click at some points on the workspace to insert some vertices, and then click on the first vertex again once you're done, to complete the polygon.

And finally, when we first open up a geometry page, we'll notice that there aren't many tools available, but if we click on more, it opens up more tools.

And if we scroll down and then click more again, it'll put more tools.

But we can also find the transform section here in the toolbar, which we're gonna use in today's lesson.

Here is how to rotate an object using GeoGebra.

To begin with, under your tools, scroll down until you find a section that says transform.

We wanna click on the tool that says rotate around a point, click on the object, then click on where we want to place our centre of rotation.

And then we can enter an angle and choose either counterclockwise, which is anti-clockwise or clockwise.

Let's do 45 degrees clockwise.

Let's see that again, step by step.

First, scroll down to the transform tools.

Click on the rotate around a point tool, click somewhere on the object and then click a point somewhere in the workspace for your centre of rotation.

Then we need to choose an angle to rotate.

Type in an angle and choose either counterclockwise or clockwise and then press okay.

Now remember that counterclockwise is a word used in the USA to mean anti-clockwise.

Let's now look at how to enlarge an object using GeoGebra.

Once again, under tools, scroll down until you get to the section that says transform.

We wanna select a tool that says dilate from a point, with dilate being another word for the transformation enlargement.

Click somewhere on the object, then click a point on the workspace where we want to insert the centre of enlargement.

I'm gonna put it here.

It then says factor.

We type in our scale factor here, I'm gonna do a scale factor of two.

Let's go through that again, step by step, click on the dilate from a 0.

2, and then click on the object somewhere.

Click a point in the workspace for your centre enlargement.

Then type in what you want your scale factor to be and press okay, and that creates our image.

Remember, in the USA, the word dilate is used instead of enlarge to mean the transformation that causes a change in size.

One of the advantages of doing rotations and enlargements on dynamic software rather than paper is that the centre point can be moved after the image has been created.

Let's check what we've learned, in GeoGebra, what button allows you to perform a transformation that causes a change in size? Is it A, dilate from a point, B, enlarge from a point, or C shrink from a point? Pause the video, make a decision, and press play when you're ready for the answer, the answer is A, dilate from a point.

What word is used in the USA for anti-clockwise? Pause the video, write down the word, and press play when you're ready for the answer.

The USA used the word counterclockwise for what we say anti-clockwise.

Okay, over to you now for task A, this task contains four questions and you'll need to have access to a web browser and the links are contained in this slide.

For question one, click on the link that says Task A Question 1, and it'll open up a GeoGebra activity.

In that question, you need to enlarge object A onto image A prime, and write down what is the scale factor you used and what is the coordinate of the centre of enlargement that you used.

In question two, open the link that says Task A Question 2.

And here you need to enlarge object B onto image B prime.

And again, write down the scale factor and write down the coordinates for the centre of enlargement, pause the video, have a go at those questions and press play when you're ready for the next questions.

Here are questions three and four.

In question three, click on the link Task A Question 3, and we need to rotate object C onto image C prime.

When you've done that, write down what is the angle of rotation that you used and what are the coordinates at the centre of rotation.

And in question four, click on the link, Task A Question 4.

And this one, we need to use the slider to rotate object D onto image D prime, and then write down what is the angle of rotation that maps object D perfectly onto image D prime.

Pause the video, have a go at these and press play when you're ready to work through it together.

Okay, well done.

Let's now go through Task A together.

Question one looks like this, it said enlarge object A onto image A prime.

To do that, I need to first figure out what the scale factor is.

I can see that the base of object A is two long and the base of the image A prime is eight long, so the scale factor must be four.

Let's now enlarge the shape.

Now I don't know where the centre of enlargement is just yet, but I'm gonna click on any point to begin with and I'll move it afterwards until I find the right place.

You can see now that the centre of enlargement isn't quite the right place because my image isn't quite overlapping where it should be.

Let's now move this around to see where it should end up.

The centre of enlargement is at the point (2,6) and the scale factor is four.

Question two looks like this.

Enlarge object B onto image B prime.

Well to begin with, I wanna see what a scale factor is.

I can see that B prime, the image is smaller than the object, and I can see it's got a base of two and the object has a base of four, therefore the scale factor must have been a half or not 0.

5.

Let's now do the enlargement.

Once again, I don't know where the centre enlargement is, but I'm guessing it's somewhere over here.

Hmm, it's not quite in the right place there.

So let's now move the centre of enlargement around until I get it overlaying perfectly.

The centre enlargement is at the point (2,4) and the scale factor is one half.

Okay, now we've seen it done.

Here's a reminder of the answers, in question one, the scale factor was four and the coordinate of the centre enlargement is (2,6), and in question two, the scale factor was a half or not 0.

5, and the coordinate of the centre of enlargement is two four.

Question three look like this.

Rotate object C onto image C, to begin with, I can see that it is a rotation of 90 degrees clockwise or 270 degrees anti-clockwise.

So let's do the rotation.

I don't know what a centre rotation is yet, so I'm gonna click on a point somewhere in between the image and object to begin with and I'll move it afterwards.

So it's not in the right place at the moment.

Let's move that centre rotation round until I find it in the right place.

The centre rotation must be at the point (6,6), and the angle of rotation is 90 degrees clockwise or 207 degrees anti-clockwise.

Question four looks like this.

It says use a slider to rotate the object D onto image D prime.

And we need to figure out what is the angle of rotation between the object and image.

It's 130 degrees.

Question three, again, the angle of rotation was 90 degrees clockwise or 270 degrees, and the coordinate of the centre rotation was (2,6).

And in question four, the angle of rotation was 130 degrees.

Great work so far.

We're now on to learn cycle two, we'll be learning how to use GeoGebra for reflections and translations.

Now let's look at how to reflect an object using GeoGebra.

To begin with, I want to insert a line to use as my line of reflection or my line of symmetry.

So under tools, find the line tool, click on two points in the workspace to insert a line going through those two points, and now I'm ready to reflect, under tools, scroll down again to the transform section.

We want the tool that says reflect about a line, click on the object and click on the line to reflect over it.

Let's go through that again, step by step.

First we need to find and click on the line tool and then click two points somewhere on the workspace to draw a line through those two points and then click on the reflect about a line tool.

Click on the object and click on the line of reflection to create our image.

Let's check what we've learned there.

Which of these tools did the example use to reflect an object? Was it A, reflect about a point, B, reflect about a line, or C, reflect about a circle, pause the video, make a decision and press play when you're ready for the answer.

The answer is B, reflect about a line.

Let's now translate an object using GeoGebra.

To begin with, I want to insert a vector to describe the movement that I will translate my object with.

So under tools, find a section that says lines, and we want the tool that says vector.

Click on a point in the workspace to insert the start of your vector.

And another point for where you want the vector to end I'm gonna go three squares to the right and one square down.

This vector now describes the movement that is three squares to the right and one square down.

So when I translate my object using that vector, it will translate it three squares right, and one square down.

So under tools, find transform, we want a tool that says, translate by a vector.

Click on the object and click on the vector.

Let's do that again, step by step.

First find and click the vector tool.

Click on two points somewhere on the workspace to draw your vector and then click on the translate by vector tool.

Click on the object and then click on the vector, and that will create your image, which has been translated.

An advantage of doing reflections and translations on dynamic software is the lines and vectors can be manipulated after the image has been created.

So for example, here we've got a reflection that has been done and after I've done it, I can move the points that line round, to move the image.

Let's check what we've learned.

In addition to the translation tool, which other tool did the example use to translate an object? Was it A, the segment tool, B, the line tool, or C, the vector tool.

Pause, make a decision and press play when you're ready for the answer.

The answer is C.

The example used a vector tool to first draw a vector and then translated the object using that vector.

Over to you now for task B, and once again, you'll need a web browser and access to these links.

In question one, click on the link task B, question one and complete the follow instructions, first, draw a line through those two points.

You can see the P and Q.

Once you've done that, reflect object A over that line.

And then once you've done that reflection, the reflection won't be in the right place so you want to move that line of reflection down until the object A is on top of image A prime and write down which of those lines is the right line of reflection.

In question two, click on the link task B, question two.

You'll first need to reflect that object B over the line, and then you'll notice on the line as a point mark P, move that point onto each of the numbered points to turn the line of reflection, write down at which point does the line reflect object B perfectly onto image B prime.

Pause the video, have a go at these, press play when you're ready for the next questions.

Question three, click on the link task B, question three, and it'll bring up a page with objects, A to E, all around the page, and vectors one to nine on the right hand side of the page, you need to translate each of those objects A to E using vectors one to nine, and translate it so that the object goes onto the image.

And for each one, write down which vector successfully translates the object onto the image.

For each one of those questions, just write down the vector number one to nine in those blank spaces.

Pause the video, have a go at this and press play when you're ready to go through it together.

Well done, let's go through with some answers for task B.

Question one looks like this and it says draw a line through the points P and Q.

It then says reflect object A over line.

You can see it's created this image here which is not overlapping the image A prime down here.

So it says move the line reflection down onto each of the numbered lines and which line reflects object A onto image A prime.

Now, because it's so much higher here, I might skip a few lines to begin with.

I might move it all the way down to line five.

Now I can see that's gone too low, so let's move it up until I get to the right place.

I can see line four is correct.

Question two looks like this.

It says reflect object B over the line, and then it says move point P onto each of the numbered points to turn the line.

At which point does the line reflect object B onto image B? It's at point 6.

Okay, here's a reminder of the answers.

It was line four and in question two it was point 6.

Question three looks like this.

It says translate each of the objects A to E using vectors one to nine, which vector translates each object onto the image.

So object A is here, I wanna move it to the right and it moves right and down a little bit.

So I'm gonna first try and use vector six, that's correct.

So vector six translates object A onto the image.

Let's now do object B, that moves right and down.

I could first use this one vector one.

I can see it's not quite the right place.

So let's undo and let's try it again with vector four.

That one has done it correctly.

So vector four translates object B onto the image.

Let's try object C.

That has moved left and down.

I can see I've got seven and nine both move left and down.

Let's try seven first.

That's not quite going to the right place, so let's undo that and try again with vector nine, I can see vector nine translates object C onto the image.

Now object D, that moves right and up and it moves quite a bit of a distance.

So vector eight is correct, translates object D onto the image.

And finally, object E that moves left and up.

I could try vector one or two to begin with, sorry, that's not enough.

Let's try again.

Let's try vector three.

That one's correct.

Vector three translates object E onto the image.

And here a reminder of the answers for question three.

In three A, it was vector six.

Three B, it was vector four.

Three C, it was vector nine.

Three D, it was vector eight and three E, it was vector three.

Well done, fantastic work so far, onto the final learn cycle for today's lesson, we'll be using GeoGebra to identify transformations.

Here's an example of a set of problems where we need to find the transformation that maps object A onto images B, C, D, and E.

Each one's got a different transformation and we need to describe the transformation fully.

At the bottom of the slide, there's a link to an interactive version of this task.

Let's find and describe the transformations that map object A onto images B, C, D, and E.

And the way we're gonna do that is by performing the transformations on GeoGebra.

And we won't necessarily get the details right first time, but the benefit of using dynamic software is we can adjust those details afterwards until we've got it right.

So object A onto image B looks like a reflection.

So let's draw a line.

Now, let's reflect object A over that line.

Great, it's nearly right, but it's not quite in the right position.

So let's move that line of symmetry until it's in the correct position, there.

Object A has been reflected over the Y axis.

Now let's look at object A onto image C.

It's a change in size, so it's gonna be an enlargement, so I'll need my dilate tool.

And the scale factor, it's gone from a width of two to a width of one.

So the scale factor must be a half or not 0.

5.

Let's click somewhere around here for the centre enlargement.

Great, it's the right size, it's just not quite in the right place.

Let's move that centre enlargement around until it's in the right place, there.

Object A has been enlarged by a scale factor of not 0.

5 from the point minus four minus five.

Let's now look at object A onto image D.

It hasn't turned or flipped or changed size, so it must be a translation.

So let's first draw a vector.

The vector it needs to move it to the right and down.

Let's go there.

Now let's translate the shape.

Brilliant, it's pretty much in the right place.

It's just a little bit off.

So let's move the end of my vector.

Excellent, so object A has been translated by a vector of three minus five.

And finally, let's rotate object A to get to image E, I'm gonna click somewhere around the middle-ish for my centre of rotation.

Let's try here.

It's in the right orientation, but it's not quite the right place.

Let's move that centre rotation round.

There we are.

So object A has been rotated 180 degrees around the point (1,1).

Let's check what we've learned there.

Before we go any further, which transformation has occurred in this situation? Is it A, enlargement, B, a reflection, C, rotation or D, a translation? Pause, write your answer and press play when you're ready for it.

The answer is, B, a reflection has occurred.

What transformation has occurred this time? Is it A, enlargement B, reflection, C, rotation or D translation? Pause, write your answer.

Press play when you're ready for the answer.

The answer is D.

It's a translation, it hasn't turned or flipped in any kinda way or changed size, it's just moved down.

Which transformation has occurred this time? A, enlargement, B, reflection, C, rotation or D translation.

Pause, write your answer and press play when you're ready for the solution.

The answer is A, it's an enlargement.

We can tell that 'cause the size has changed.

Okay, over to you now for task C, click on the link that says Task C and it brings up a page with an object A, and four images, B, C, D, and E.

You need to transform object A onto each of those B, C, D, and E.

And for each one, write the description of the transformation in full.

Pause the video, have a go at this and press play when you're ready to go through the solutions.

Let's do this together now.

So we need to find and describe each of the transformations from A onto images, B, C, D, and E.

So let's look at A to B first, it's turned.

So this one's gonna be a rotation.

Let's find our rotate tool.

It looks like it's rotated 90 degrees clockwise, but let's just first click on anywhere for my centre of rotation.

It's not in the right place, but we can move the centre until we find it is in the right place.

Great, so it's rotated night degrees clockwise around the point (0,1).

Let's now do A to C.

A to C, it hasn't turned or hasn't reflected or got bigger.

It's just moved, so it's gonna be a translation.

Let's first draw a vector.

It's translated to the right and also down a bit as well.

Let's translate object A by this vector.

Oh, it's not quite the right place.

It's moved down enough but hasn't moved far enough to the right.

Let's move it.

Great, object A has been translated by the vector nine minus one.

Let's now do A to D.

There's a change in size there, so it is gonna be an enlargement.

Let's use a dilation tool.

Let's click anywhere for a centre of enlargement.

And I can see that the scale factor must be three 'cause this length is three and that other length in the object is one.

Now the centre of enlargement is not quite in the right place.

Let's move it until it is, great.

The centre of enlargement is at minus six six.

It's an enlargement by a scale factor of three from minus six six.

And finally A to E.

We can see it's a reflection and it looks like it's been reflected on a horizontal line.

It's this axis here, so it's been reflected over the X axis.

And here's a reminder of the final solutions.

Question one, A to B was rotated 90 degrees clockwise around zero one.

Question two, A to C was translated nine minus one.

Question three, A to D was reflected over the X axis.

And question four, A to E was enlarged by a scale factor of three from the point minus six, six.

Brilliant work today, a big well done.

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

Firstly, dynamic geometry software such as GeoGebra in today's lesson can be used to quickly draw shapes and transform them.

Dynamic software also allows us to manipulate aspects of a diagram after a transformation has already taken place.

So for example, if we get it wrong first time on GeoGebra, we can adjust things to get it right.

And then finally, we've also seen how sliders can be used to make values such as angles and scale factors dynamic we saw particularly with the rotation activity.

Great job today, wonderful.