Lesson video

Hello, I'm Mr. Gratton and thank you so much for joining me for today's lesson.

In today's lesson on transformations, we'll be looking at what information is required to fully define a rotation.

Well, one of the most important pieces of information needed is the centre of rotation.

Our main focus of the lesson is on describing rotations whilst referencing this centre of rotation with a further part of the lesson on finding centres of rotation for specific types of rotation.

Let's have a look at some examples of rotation.

Even if an object is rotated by the same amount, 90 degrees in this case, notice how it's possible for the image to end up in a different location time after time.

Rotations will end up in different locations depending on where its centre of rotation is.

I've referenced this phrase, centre of rotation, but what does it actually mean? It is a fixed point or pivot that dictates the distance an object rotates after a specific size of turn.

So 90 degrees, 180 degrees.

Imagine the centre of rotation as a pole in the ground, and the object is attached to this pole by a piece of rope.

If the centre of rotation is close to the object, then it won't travel a far distance.

But imagine that same centre of rotation is far away from the object.

It will rotate by a bigger amount.

So depending on how far away the centre of rotation is from the object, that will dictate how far the object will rotate to make its image.

Here's a visual demonstration of how this works.

The centre of rotation has stayed fixed like the pole in the ground and the object is what rotates and it always stays the same distance away from that pivot, from the centre of rotation.

Now the centre of rotation can be anywhere.

It can be on the vertex of the object, it can be outside the object, like the example I just gave you, or it could be inside the object where there is always going to be a lot of overlapping between the object and its image.

Okay, so that's what a centre of rotation is, but how can we describe it in practical and mathematical terms? How? Well, as a coordinate, this is the easiest and most mathematical way.

In this diagram, the centre of rotation is one single point where the object and the image intersect.

Furthermore, this single point is a corresponding vertex between the object and the image.

If the centre of rotation were to be on a shared vertex of the object and image, it will always be a pair of overlapping corresponding vertices and never a random pair of vertices that just so happen to lie on top of each other.

So centres of rotation are an important part of describing a rotation, but it's not the only part that we need to use.

There are in total four pieces of information you need to describe to fully describe rotation.

The part people tend to forget, or simply not put as much focus on, is simply saying this is a rotation.

So say it is a rotation is part one of the description.

The direction, so clockwise or anti-clockwise, that is part two.

The size in degrees, so 90 degrees, 180 degrees, that is part three, and then the centre of rotation is part four.

So in this diagram, let's look at the four pieces of information.

Saying it's rotation, check, part one.

We are rotating it anti-clockwise, we know that from the arrow that's helping us visualise the rotation.

We know it is rotated by 90 degrees and the centre of rotation is again one single point where the object and the image intersects.

And that point is the coordinate seven, three, and that is the centre of rotation.

So to say it as one sentence, we could say the object has been rotated anti-clockwise through 90 degrees, about seven, three.

Okay, check time for you all.

Which of these is the correct coordinate for the centre of rotation for this object? Pause the video and have a think at what coordinate it could be.

Three, six.

Notice how three, six is the only point where the object and the image overlap and therefore that is the centre of rotation.

It is also marked by that cross, but do not rely on it 'cause that cross may not always be there.

Next check.

I've given you a few parts of the description for the rotation.

What is the missing part of the description? Have a think about what's missing.

Pause the video and off you go.

Again, the piece of information that's missing is the coordinate for the centre of rotation and the coordinate for the centre of rotation is two, four.

Well done if you spotted that.

Here's an example of when I haven't given you that cross to plot the point of the centre of rotation, what is the centre of rotation? That one missing piece of information about the rotation.

Pause the video and have a go.

And the centre of rotation is the coordinate seven, two.

Well done if you've got that right.

This time it is a true or false question.

If a vertex of an object is the centre of rotation, then the location of the vertex is invariant.

Think about what the word invariant means.

Answer true or false and look at which description justifies your answer.

Pause the video and work through what you think the answer is.

So the answer is true.

Invariant means a property of a shape that has not changed.

If a vertex of an object is the centre of rotation, that vertex does not move.

That vertex, that point on the shape is the pivot and the pivot never moves when an object is being rotated.

So we have looked at examples where the centre of rotation is on the vertex of an object, but the centre of rotation can also be outside the object.

Nearby or far away, there is no limit to where the centre of rotation can be.

When we're dealing with the centre of rotation outside the object, imagine that the object is attached to the centre by a rope.

You can physically draw that on any diagram that you have with a centre of rotation outside of the object.

Sometimes that helps you visualise how much the rotation has rotated by.

So check time.

Complete Izzy's description of this rotation.

The object has been rotated 90 degrees in which direction and with what coordinate as the centre of rotation? Notice the details on the diagram, the arrows and the coordinates that have been plotted to finish off your answer.

Pause the video and fill in all the blanks.

So we know that it is a clockwise rotation because that arrow is showing a clockwise turn.

The point that is outside the object and the image is clearly the centre of rotation.

So all you need to do is give its coordinate, and the coordinate is six, two.

Okay, let's have a look at a couple of misconceptions.

Alex says that if the centre of rotation is outside the object, then the object and the image will not overlap after a rotation.

You might think this is true because of examples like this.

You've got the object and the image rotated 180 degrees and that centre of rotation is at five, seven.

The object and the image do not overlap.

But what if we took the same object, the same centre of rotation, but rotated it like this instead by a 90-degree clockwise turn? Well, we can see that the object and the image now do overlap.

It is possible.

It is even likely at times that the object and the image will overlap after a rotation.

So do not let the fact that the centre of rotation is outside the object trick you into thinking the image will never overlap its object.

Okay, we have dealt with centres of rotation being on the vertex of an object.

We've dealt with them outside the object, too.

Centres of rotation can also be inside the object.

When the centre is inside the object, then there is at least a slight bit of overlap, sometimes mostly an overlap between the object and the image.

Here's an example.

Notice how at different amounts of rotation, there is different amounts of overlap, but there is always some overlap.

Okay, check for understanding time.

Complete Laura's description of this rotation.

The object has been rotated by 90 degrees in the which direction and this time I've not given you an arrow to guide you.

Furthermore, what is the centre of rotation? Pause the video and think through all of those details.

So by starting with the object, we've rotated 90 degrees anti-clockwise to get the image and the centre of rotation is still marked on the diagram as the coordinate five, three.

Right, onto some independent practise.

The diagram on screen shows a horizontal object and a vertical image.

It has been rotated anti-clockwise.

Fill in the missing information to show how much it has rotated by and what the centre of rotation is.

Pause the video to fill in those details.

Question two is very similar, but this time I have given you diagonal lines rather than horizontal and vertical ones.

Regardless, we know that the object is rotating clockwise.

Figure out by how many degrees and what the centre of rotation is.

Pause the video to complete the full description of this rotation.

I've done away with the lines.

Now we have two shapes, an object and an image, both trapezia.

In this diagram, we spot that the centre of rotation is outside the two shapes.

I have given you those guidelines, the rope that is attached to the centre of rotation, the pole in the ground.

This will help you figure out how much the rotation has rotated by in degrees.

Complete the description by filling in the rotation amount and the centre of rotation.

Pause the video and give it a go.

Question number four, very similar.

I've given you an object and an image attached to the centre of rotation by those lines.

Complete the description for the rotation by filling in the number of degrees it has rotated by and the centre of rotation as a coordinate.

Pause the video and off you go.

Right, question number five is slightly different.

I've given you an object and an image, but this time I've given you two descriptions.

One description talks about a clockwise rotation whilst the other is an anti-clockwise description.

Fill in all the missing details for both descriptions.

Pause the video and spot any patterns in the two descriptions that you might have.

Question number six, similar again, I've given you two descriptions, but this time I've given you less detail.

Fully describe the rotation in two different ways and be careful about the detail.

Pause the video and good luck.

And finally, question number seven.

This time I have the object and the image overlapping quite a lot.

There are multiple possible descriptions that could correctly describe this rotation.

Pick one way and fully describe in that way.

Pause the video for this final question of task A.

Okay, onto the answers.

Question number one, we can see that the object and the image lines are 90 degrees apart.

Therefore, it is a 90-degree anti-clockwise rotation.

We can tell where the centre of rotation is because it is the one coordinate where the object and the image intercept.

That is the coordinate eight, six.

For question number two, we can again see that the object and the image are 90 degrees apart, but this time, because it states a clockwise rotation, you cannot use that short rotation from the object to the image.

You have to go the long way around clockwise.

That is why it is a 270-degree rotation.

The large angle on the outside clockwise, not on the inside, anti-clockwise.

Again, the centre of rotation is at three, five, the one point where the object and the image intersect.

Here's my advice for this sort of question.

You did not need to look at the entirety of the object and the image.

That line that connects the centre to the object and the centre to the image is all you need to look at and we can observe that here, yet again, there is a 90-degree rotation.

We know it's 90 and not 270 because the arrow indicates a clockwise rotation.

As the centre of rotation is marked, we can just say it is the coordinate eight, seven.

Question number four, similar to question number three.

We know it is a 90-degree rotation because of the anti-clockwise nature of that rotation.

It is not the long way around.

And again, the centre is at the coordinate eight, eight.

Onto question number five.

For the clockwise rotation, it is 180 degrees and for the anti-clockwise direction, oh, it's also 180 degrees.

Remember, 180 degrees is the same result no matter whether it is clockwise or anti-clockwise.

Furthermore, no matter which direction we are rotating an object, the centre is the same.

So regardless of clockwise or anti-clockwise, the centre of rotation is always eight, six.

However, there are more possible answers that some of you might have spotted.

180 degrees is the most obvious common answer, but you could also have a rotation of 180 degrees plus 360, which is 540 or 180 degrees plus two lots of 360, that's 900.

This is because adding 360 degrees or multiples of 360 degrees does not change your answer because a full turn of 360 degrees maintains the orientation and position of any object.

Okay, question number six.

For the clockwise direction, we can see that it has a 90-degree rotation.

For the anti-clockwise direction, we know it's gonna be the rest of 360 degrees, a full turn, and so that is going to be 270 degrees.

Again, regardless of direction, the centre of rotation is always gonna be the same and the centre of rotation is always going to be the coordinate eight, four.

And finally, question number seven.

Because the object and the image are so perfectly aligned horizontally and vertically, we can see that the centre of rotation is slap bang in the dead centre of the two shapes and the centre of the two shapes is the coordinate five, six.

Now one answer you could give is that it has been rotated 45 degrees, either clockwise or anti-clockwise, but actually there are infinitely many answers and each answer is going to be either 45 degrees or 90 bigger than 45 degrees or 90 degrees bigger than the previous answer.

So 45, 135, 215, 305, 395 are all valid answers to this rotation.

As long as you have the centre at five, six, the degrees with which it has turned can take on many values.

For all the examples of rotation we've seen so far, the centre of rotation has been either given to you or very obvious as to where it is, possibly because it is on the vertex of an object.

Sometimes it is not as obvious, but we can still figure it out.

We'll be looking at today finding the centre of rotation for 180-degree rotations where the centre of rotation is outside the object and image.

Let's have a look.

In this diagram, we can see that the object and the image are very far apart.

This means that the centre of rotation is going to be outside the two shapes.

If the two shapes overlapped or the vertices touched, we would know that the centre of rotation was inside or on the shape.

But because they are far apart, we know that the centre of rotation is outside.

We also know that the rotation is a 180-degree rotation, so we know that the centre of rotation is gonna be outside the two shapes, but where exactly? We can go through a few simple steps to find out that centre of rotation for 180-degree rotations.

Step one is to connect together corresponding points on the object and an image with a line segment.

And after we've drawn at least two line segments, we can find where the line segments intercept.

So here's how you do it.

I've connected those two corresponding points with a line segment, and then I've connected a separate pair of corresponding points with a different line segment.

The location where those two points intersect is the coordinate six, five, and we can say that the centre of that rotation is the coordinate six, five.

Okay, some practise time to check your understanding of what corresponding points are.

The object is at the top of this diagram, and I have labelled for you the point A.

What is the corresponding point to A on the image at the bottom of the shape, E, F, G, or H? Pause the video and think about your answer.

The answer is point H.

Second question.

Let's use this diagram.

I have connected together A with its corresponding point on the image.

By mentally connecting together a second pair of corresponding points, and there is one really obvious one to pick.

Find where the centre of rotation is.

Pause the video and think through it.

So the pair of corresponding points that I recommend connecting is this pair, which leads to the answer of four, five.

The centre of rotation is the coordinate four, five.

This could also work with the same centre of rotation found no matter which pair of corresponding points you've chosen.

However, it only works if they are corresponding points and not just some random pair of points, one of the object, one of the image.

And for task B, the final set of tasks for you to do.

Below are two rotations where an object and its image have two pairs of corresponding vertices labelled.

So A and A prime are a pair of corresponding vertices.

B and B prime are also a pair of corresponding vertices.

By connecting together with a line segment all corresponding points, find the centre of rotation of each pair of objects.

Pause the video to connect those corresponding points with a line segment to find each centre of rotation.

Question two is very similar to question one, but I have not labelled the corresponding points.

You have to figure out what points are corresponding with each other yourself.

Besides that, it is the same task.

Connect the corresponding points together with the line segments to find the centres of rotation.

Furthermore, for question three, write down the one answer to when we can find centres of rotation using this method.

Pause the video and have a look at those two questions.

Onto question number four.

Yes, you still have to find the centre of rotation, but this time I want more.

I want you to fully describe the rotations between the two objects and their corresponding images.

And remember, every full description involves four parts.

Pause the video, remind yourself about what those four parts of the descriptions are, and good luck.

Okay, onto the answers.

For question number one, the two centres of rotation are six, six and 17, five.

Well done if you drew on those line segments and you connected them to the correct corresponding points and extra well done if you got the correct coordinates for the centres of rotation.

For question number two, the centres of rotation are three, four, 13, five, and 22, four.

And we know that this method only works if the rotation is a 180-degree rotation.

Question number four, object one has been rotated by 180 degrees.

Those are already two parts of your description, rotation, and 180 degrees.

The centre of rotation is six, four.

Now I'm missing the fourth part of the description because anti-clockwise and clockwise are both equally correct for this question.

Object two has been rotated again by 180 degrees, but this time around the centre of rotation 15, four.

Well done if you labelled on those line segments that connect corresponding points and extra well done if you found the centres of rotation and wrote down full descriptions of them.

Well done in everything that you've done in today's lesson.

In today's lesson, we have covered all the features needed to fully describe a rotation.

This is the direction, the number of degrees, and the centre of rotation, which is a very specific feature for rotations only.

We have also looked at how to find the centre of rotation when we have an object and an image that have been rotated 180 degrees.

Thank you so much again for all your hard work, and I hope to see you again in another maths lesson.

Have a good rest of your day.