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 looking forward to the lesson and willing to try your best.

So, our outcome for today's lesson is to be able to describe and perform a rotation of an object.

On the screen, there are some keywords that you've met previously in your studies.

It may be a good idea for you to pause the video here and just reread them to make sure you're familiar before we make use of them during this lesson.

Press play when you're ready to move on.

So, in today's lesson of checking and securing understanding of rotation, the lesson is gonna be in two learning cycles.

The first one, we're gonna look at how to perform a rotation, and in the second learning cycle, we're gonna look at describing a rotation.

So, let's make a start at performing a rotation.

So, using a rectangular sheet of paper so if you haven't got one nearby, go and find one.

So, a rectangular sheet of paper doesn't have to be too large.

Line it flat on a surface and using your finger.

So just use probably your pointing finger as a pivot point, rotate the piece of paper.

So, place your piece of your finger; it can be anywhere on the sheet of paper and rotate it.

So, I've done it sort of quite central and rotated it, and now my piece of paper is in this orientation.

If you would now go back to the original orientation of your sheet of paper, how you had it first up, but move your pivot point, move where you are placing your finger, would the outcome be the same? So have a go.

So, this is what happened to mine.

So, are the outcomes the same? Well, the orientation might be the same.

So, if we talk about words that we use with pieces of paper, I had it originally in sort of landscape position.

And when I rotated it, when I twisted the sheet of paper, it finished in what we call portrait.

So, both of them ended as portrait.

So, they might end up with the same orientation, but the position is different.

And you can see that relative to my original starting place with the dotted lines.

So how about we do it again? We do it twice again.

So, I've gone back to my sort of landscape.

But this time, we're not gonnamove the finger.

The pivot point will stay in the same position in both attempts, so how else could we produce a different outcome? So last time, our position of the piece of paper was different, so how can we find a different outcome without moving our finger? Well, we could rotate the paper a different amount, a different angle.

So here again, the first one, I've rotated, and I've gone from landscape to vertical, whereas on the second one, I've rotated, and I've not quite gotten through to the vertical position.

So, any shape we use using a rectangle there because it's a rectangular sheet of paper.

Any shape can be rotated, and its image will land in various positions depending on the centre of rotation and the amount.

And we can see that there.

So, the orientation of the object changes when it is rotated.

So, on the most part, the orientation will have changed when a rotation has taken place.

So, to perform a rotation, it is useful to have a piece of tracing paper, and that will help us to see where the rotation finishes.

So here we've got a centre of rotation marked with the X, so that would be where you had your finger pointing on your sheet of paper, and we've got our object, let's say that's the purple one, and our image is the pink one.

A rotation has taken place, so the orientation of the triangles are different.

If I rotate it a little bit further, the orientation has changed and the orientation has changed.

So, let's look at how we perform a rotation.

So rotate shape A.

So that's our object, the original shape, 90 degrees clockwise about the origin.

So, we've got a shape A on a coordinate axis, and we need to rotate it.

So, the first step will be to mark the centre of rotation.

For some questions, the centre of rotation may already be indicated, it may not be on a coordinate axis, it just may be marked with a cross or a dot, it will tell you within the question.

But if not, you need to mark it.

So, the origin is a specific pair of coordinates, which is 0, 0.

So we mark the centre of rotation.

Now we're gonnamake use of the tracing paper.

So, we lay our tracing paper over the top so that the object and the centre of rotation are both within the tracing paper.

And then you're going to sketch the centre.

So, on top of your tracing paper, draw the centre point and the object.

We don't have to use a pencil and ruler for this, but you do want to try and be fairly neat so that your shape doesn't change when we rotate.

A tip is to draw an arrow, a vertical arrow from the centre point onto the tracing paper.

We'll come back to this in a moment, so draw us.

That is an optional choice; you don't have to do that.

But by drawing a vertical arrow, I think it can be really helpful to know when to stop turning the tracing paper.

So step three is to use your pencil as your pivot point.

So, that was your finger when we first started, and rotate the tracing paper the given amount in the question and in the correct direction.

So our question is 90 degrees clockwise.

So clockwise is sort of turning to the right.

So arrows indicating the direction we need to turn, the same way that normal clock hands would twist.

And this is where my arrow tip comes in.

So we need to turn it 90 degrees.

Well, 90 degrees is one-quarter turn of a full turn.

So if we think about the direction of the arrow currently, the arrow, and we could use compass directions here, the arrow is pointing north.

So if you were to turn that one-quarter to the right, it would face east.

It's currently pointing up.

If we turn it one-quarter to the right or 90 degrees, it will face to the right.

And this is where our tracing paper needs to finish after our rotation.

So here we now can see where the image is located.

So, we now need to get the image from our tracing paper back onto our sheet of paper.

So, step four is there's two options.

Either you can push down on each vertex of the image to leave a little dent in your sheet of paper, or you could carefully lift your edges of your tracing paper to mark onto your worksheet.

(keypads clicking) So here we can now see our four vertices, it was a quadrilateral, so there are four, and we can remove the tracing paper and draw on the image, and your rotation is complete.

So, the tracing paper is a really useful aid there to know where the image ends up being after the rotation.

So, here's a check.

After placing the tracing paper over the paper, that's your worksheet or your exercise book, what two things must you draw onto your tracing paper? Pause the video whilst you think about the two things you need.

It might be that you go back and re-watch a bit of the video, and then press play when you're ready to check.

So, the two things were the object, so that's the original shape and also the centre of rotation.

May have been that you thought about that arrow as well.

The arrow is optional.

Okay, so the vertical arrow is not something you have to do, but I think it's a really useful way of knowing how far to turn your tracing paper.

A second check.

The object needs to be rotated 90 degrees clockwise around the centre of rotation marked.

Which of these is the correct placement of the tracing paper after that rotation has happened? So, pause the video, have a look at A, B, and C, and think about which one has ended up in the correct position.

Press play when you're ready to check.

So, the answer is B.

On A, it's turned 180 degrees, so it has turned too far, and on C, it's turned to the correct amount in the correct direction, but it hasn't been pivoted around the centre of rotation; they didn't put their pencil on the centre of rotation and twist about that point.

Another check: after which step do you push your pencil firmly onto each vertex of the shape on the tracing paper to leave an imprint on the paper A, B, or C? So pause the video, read the options, and then decide which one is the correct answer.

Press play when you're ready to move on.

So, you're going to press down on each vertex after the rotation has taken place.

So here we have Andeep and Jun, and they've both attempted to rotate shape a which is a trapezium, 90 degrees anticlockwise about the pair of coordinates 0, 4.

However, they've got different results.

Have either of them done the rotation correctly? So have a moment.

Pause the video if you need to.

And which one, if either, have done it correctly? Well, actually, neither of them have done it correctly.

And there's two different reasons.

So, Andy has used the wrong centre.

So, he has plotted the coordinate 4, 0 when he should have plotted 0, 4.

So be very careful, making sure you are plotting your centre of rotation accurately, whereas Jun has plotted the correct centre of rotation, but turned or rotated in the wrong direction.

Jun rotated clockwise where the question wanted anti-clockwise.

Here's a check for you.

Shape A has been rotated 90 degrees clockwise about negative 2, 2 which is the image.

So, you can see shape A there.

So, is it B, C, or D? Pause the video whilst you make a decision, and press play to check your answer.

B is the image 90 degrees clockwise, so that would be turning to the right one-quarter turn lands at B, so it's the first task of the lesson, which is for performing a rotation.

So on question one, there are two parts for you to do.

The centre of rotation has already been marked onto the coordinate axes.

You can see that with the little cross and you need to rotate 90 degrees clockwise for part A and rotate 180 degrees clockwise for part B.

Pause the video whilst you complete those rotations.

And then, when you press play, we move to question two.

Question two, more performing of rotations necessary, but part A and part B, there is now the centre of rotation within the question.

So, you need to also plot that coordinate before you start.

So, pause the video whilst you are working on A and B.

When you press play, we're gonna move to question three.

So last question of task A is question three.

So, the rotations have taken place.

However, there are some errors.

So, you need to identify the error in each case and then correctly perform the rotation.

So, press pause whilst you are doing both things.

And then, when you're ready to go through the answers to task A, press play.

Here is questions one's answer.

So, you needed to rotate 90 degrees clockwise on part A, and therefore your image should be in this location.

I'm gonna read out the coordinates of the vertices.

So, the first vertex, if we start at the origin, 0, 0, going anti-clockwise around the image.

The next one is 1, negative 1.

Then it's 2, 0.

Then 2, 1.

And finally, 0, 1.

Note here that the image and the object are touching on one vertex.

There is a shared vertex there to one.

On part B, you needed to rotate 180 degrees.

There wasn't any direction.

180 degrees is half turn.

So, you need to rotate it half a turn.

Again, I'm gonna read out the coordinates of each vertex on the image.

So, starting on the vertex 0, 1 and going anti-clockwise, the next vertex is negative 1, 1.

Then negative 2, 0.

Negative 1, negative 1.

And finally, 0, negative 1.

Question 2, you needed to perform the rotation, which this time included also plotting the centre of rotation.

So, on A, it was the origin, so that's the point 0, 0.

You needed to go 90 degrees anticlockwise, so have a look at where the image is located.

Once again, I'll read out the vertices.

So, starting on negative one, one.

I'm gonna go anti-clockwise around the shape.

So negative one, one, then negative one, two, then negative two, two, then negative three, three, then negative three, one.

On part B, it was a rotation of 180 degrees about negative 1, 1.

So, hopefully you plotted the centre in the correct location.

We can see the orientation of the shape is different because we've rotated 180 degrees.

And our image has landed up in the position where, starting on the vertex, that is the origin, going anticlockwise.

So, 0, 0, 0, negative 1, 1, negative 1, 1, 1.

And they are the four vertices because it is a quadrilateral.

Question three, you need to identify the errors and then also perform the transformation correctly.

So, on part A, the centre of rotation was plotted incorrectly.

They had plotted 0, 3, when the question wanted 3, 0.

So being very careful.

And then you can see where the image should be located.

The three vertices are 2, 3, 4, 2, and 5, 3.

On part B, the rotation had been the rotation happened in the wrong direction.

So, your image should be in that third quadrant, and the four vertices are negative three, negative two, negative three, negative three, negative one, negative three, and negative one, negative two.

Really well done if you got all of those correct or most of those correct.

If you didn't, make sure you go back and look at why or how, what went wrong.

Perhaps you plotted the centre incorrectly, or maybe you went in the wrong direction.

So, the second learning cycle is now about describing a rotation.

So, when a transformation has taken place, recognising a rotation and also knowing all the parts of a full description.

So, if a shape has been transformed, then we need to firstly identify the type of transformation.

So, transformations you will have covered in the past.

The four that we sort of focus on mostly are rotation, translation, reflection, and enlargement.

So, first of all, you need to identify the type of transformation.

In this lesson, they're all gonna be rotations.

If a rotation has taken place, then one way we will know this is the case is the object and image will be congruent.

Remember that congruent means that the two, the object and the image, are exactly the same size and shape.

If you could cut out both accurately, then when you overlap them, they would be exactly the same.

The image will not be a reflection if a rotation has taken place.

So that means the sense will be the same as the object, but the orientation will have changed.

So here, if this was our question, if we'd been told a transformation has taken place, we would know that it cannot be a rotation.

And why do we know that? Well, we know that because the two shapes are not congruent.

There are two right angled triangles.

So, the shape in terms of a description of the shape, they are both right angled triangles, but their size is different.

If we look at the perpendicular edges of those right-angle triangles, one of them is two units by three units, and the other is four units by two units.

So, we know that they are not congruent.

Therefore, it cannot be a rotation.

The shape has also flipped over.

The sense of the two shapes is changed.

So, it cannot be a rotation if the sense has changed.

Whereas on this one, it could be a rotation.

And so, why might this be a rotation? Well, the two shapes are congruent.

Once again, they are right angled to triangles, but this time the dimensions are the same.

They are two units by three units if we look at the perpendicular edges.

Also, they haven't flipped over, so the sense of the two shapes haven't changed, but the orientation has.

So, we can see that the orientation of these two shapes has changed.

So, a rotation might have taken place.

So, here's a check, which of these transformations might be a rotation? It might be easier to think about, you might be able to disregard some quicker because you know they cannot be a rotation.

Pause video whilst you're doing that, and when you're ready to check, press play.

So, the answer is B, and the reason it couldn't be A is because the object and image are not congruent, that lends itself to look more like an enlargement.

And it can't be C, although they are congruent, the object and image they have flipped their sense has changed.

So here on a check, when a rotation has taken place, the object and image are A always, B sometimes, or C never congruent.

Pause the video, make a decision what word fits there and then press play to check and then press play to check.

This one is always.

So, if a rotation has taken place, then the object and image will always be congruent.

A similar check here.

When a rotation has taken place, the object and image always, sometimes, or never overlap.

Pause the video.

It may be that you want to look back at some of the questions you did in task one or have a little try on a sheet of paper, and then when you're ready to check, press play.

So sometimes this is dependent on where the centre of rotation and how much you turn in your rotation.

Sometimes the object and image do actually overlap.

Again, when a rotation has taken place, the object and image always, sometimes, or never touch.

Pause the video when you're ready to check, press play.

They sometimes touch.

We saw that on one of the answers in task A, that sometimes a vertex on the object and the vertex on the image are in the same position, so they are touching.

When a rotation has taken place, the object always, sometimes, or never changes orientation.

Pause the video, think about this one carefully.

When you're ready, check press play.

If you went for always, there is one exception to the rule.

And that's why it's sometimes, and that exception to the rule is if you were to rotate something a number of full turns, then it would have the same orientation.

And this is very rare that we see this in an example or a question, because the object and image would be completely the same.

They'd be on top, they'd be in exactly the same location.

But if you think about this from yourself, from a practical point of view, if you face a particular wall, and you turn yourself one full turn, then you would still face the same wall that you started.

If you did two full turns, you would face the same wall, so your orientation wouldn't have changed despite the fact that you did rotate.

And so this is the sometimes it's an exception to the rule, but on the most part, if the orientation has changed, then a rotation is likely to have taken place.

So once you've established that the rotation has taken place because an orientation change has happened, because the object and the image are congruent, then you need to find the centre of rotation.

The centre of rotation is that pivot point, and it depends on that pivot point as well as the amount of turn for the location of the image.

The centre of rotation can actually be anywhere.

It could be on a vertex of the object.

And here, you can see that they are touching.

So that was sometimes they touch.

But it might be outside of the object.

So off of the object.

And so here, they are not touching.

Or it could be inside the object.

And here we can see that they are overlapping.

So dependent on the shape of the object, the amount of turn and where the location of the centre of rotation really does matter and determines where the image is.

There is a link there to a geod profile, that I would encourage you to pause the video and open that link.

And you can have a play around, you can move the centre of rotation, you can change the amount of turn to really look at how the object and the image are related; and the sort of position of the two of them dependent on where the centre is and how much turn.

Using a sheet of tracing paper is one way that we can locate the centre of rotation, although it might take a few attempts, so you do need to be resilient.

But if you did just have a pause and play with that GeoGebra file, then hopefully you've got a bit of a better understanding of where the location of the centre will be because of how the object and the image are relative to each other.

So, here we've got a transformation.

We can say it's a rotation because the object, and image are congruent, and the orientation has changed.

So, if we place our tracing paper over the top and I've traced the object, and I've marked a centre of rotation.

I don't know if that is the centre of rotation, but that is gonna be my first attempt at locating it.

So, I twist my sheet of tracing paper, and it hasn't landed in the correct spot.

So, we need to try a different centre of rotation, and this is where it might take multiple attempts.

However, before we quickly move the tracing paper, look at where it has landed.

It has landed around about the image, not quite on top of the image, which is what we want.

But we're not a million miles away.

So, we now are going to try a centre of rotation similar and close to the one we already tried.

So, let's try here.

And this is where it's worth noting the centre of rotations do not have to be integer values.

So, I've just moved over a half.

I've kept it on the same horizontal line.

The reason I've kept it on the same horizontal line is where the image landed on my first attempt was in line with the image.

So, I do not want it to go higher or lower.

It was in the correct horizontal line, if you like.

So, I've just moved my centre of rotation along the same horizontal line.

And this time, it has landed in the correct spot.

So, I now know my centre of rotation is 1.

50.

So, we can start writing a description of rotation because we need to state the transformation about 1,5 0.

The other part of a description is the amount of turn that has taken place and in which direction.

So once again, if we bring back our tracing paper and we use our vertical arrow, we can think about how much the vertical arrow has changed.

So if I draw a vertical arrow onto my centre of rotation on the tracing paper, and then I turn, I rotate the tracing paper, pivoted about that centre, I can see that a half turn has taken, I've twisted the tracing paper half a turn, which is equivalent to 180 degrees clockwise.

But did I have to rotate it clockwise? No.

So because it was a rotation of 180 degrees, it doesn't matter in which direction because it's a half-turn either way.

And you would have practised that on the task A, that when they asked you to perform a rotation 180 degrees, it didn't give you a direction because the direction doesn't matter.

So, here's a check for you.

Write a full description of this rotation from shape A onto shape A prime.

Use the tracing paper and the arrows to support you with that description.

Press pause whilst you're writing that full description, and then when you're ready to check, press play.

So, there are two correct answers.

You may have gone for 90 degrees clockwise or 270 degrees anti-clockwise.

So, the direction and the amount of turn are sort of a pair together.

So, you needed to say a rotation, so you needed to give the type of transformation, a rotation, so you need to give the type of transformation, then we need the angle and the direction, as well as the centre.

So, a rotation of 90 degrees clockwise about negative 1.

4, or a rotation of 270 degrees anticlockwise about negative 1.

4.

So, four pieces of information are needed to fully describe a rotation.

We need to state that it is a rotation.

We need to give the direction, the size of the turn, and the centre of rotation.

So, on this one here, the object and the image, that's been rotated.

We know that because the object and image are congruent, and the orientation has changed.

Anticlockwise is the direction that he has turned by 135 degrees.

So also, worth noting here that mostly you can think about quarter turns, 90 degrees, 180 and 270.

But of course, and you would have seen it on that GeoGebra file if you did open it and play with it, that the degree could be any degree.

It could be a decimal amount as well.

And so the rotation doesn't have to be stuck on quarter turns the centre rotation is marked, and there's four, two, they are the four pieces of information, but we should write that as a full sentence.

The object has been rotated anti-clockwise through 135 degrees about four, two.

We've stated the four pieces of information, but remember, if it was 180 degrees half a turn, then we do not need to give direction that's a special case.

Here's a check Lucas has described a rotation shape A has rotated 215 degrees, about four, five.

What piece of information has Lucas missed in his description? Pause the video, and then when you're ready to check, press play.

So, Lucas has missed the direction.

The direction really does matter because if you rotated a shape 215 degrees clockwise, the image would land in a different position than if you had done 215 degrees anti-clockwise, so it's really important that we have all four pieces of information in our descriptions.

So on to the last task, you need to complete the descriptions.

There's a missing part in A, B, and C.

Press pause whilst you fill the blanks, and then when you press play, we'll move on to questions two and three.

Here's two and question three as well.

So, on question two, you need to identify what is missing from each of these descriptions.

And on question three, you need to use the words at the bottom to fill and complete the sentence.

Pause the video whilst you're doing questions two and three, and then, when you press play, you move on to question four.

Question four, there are four rotations, and you need to fully describe each one.

So, pause the video whilst you write your full descriptions for part A, B, C, and D.

When you press play, we're gonna go through our answers.

Here's question one.

Remember, you were just filling in a blank on the description.

So, part A, you needed to state it was anti-clockwise because we had said rotated, so that's the transformation.

The size of turn or the amount of turn was stated, the centre of rotation was stated, the only thing that wasn't was the direction.

On part B, you needed to put in the size, or the amount of turn, which in this case was 90 degrees.

And for part C, you needed to complete the centre of rotation, the coordinates for that, which was 0, 1.

Question two and three.

Question two, identify what was missing from each of these descriptions.

So, on A, it feels like it's very full.

It's got the amount of turn, it's got the direction and the centre of rotation.

What it hasn't got is what it is, and that is a transformation, and that particular transformation is a rotation.

So, it's really important to state the type of transformation.

On part B, it says rotate, so that's the transformation.

It says 180 degrees, but it hasn't given us the coordinates of the centre of rotation.

For part C, it says rotate 270 degrees, about 3, 4.

So it's got rotate, which is the transformation.

It's got the amount of turn, and it's also got the centre, but it doesn't have the direction.

On part B, it didn't have the direction, but this was the special case of 180 degrees.

And finally, on part D, rotate 45 clockwise about negative 2, negative 4.

It doesn't have the unit, it doesn't have the degree symbol.

So, there are other units for amount of term.

Degrees is not the only unit of measure, so you do need to make sure you've got your symbol for the units.

On to question three, you needed to use the words below to complete this sentence.

So, to describe a rotation fully, you must state that it is a rotation.

Give the coordinates of the centre of rotation, the amount of turn, and the direction.

So, they are the four things you need in a full description.

And finally, question four, there were four descriptions of rotations necessary.

So, part A, a rotation by 90 degrees clockwise about 0, 1.

If you had written a rotation by 270 degrees anticlockwise about 0, 1, you are also correct.

So other than 180 degrees, there's always two correct descriptions.

For part B, a rotation by 180 degrees about zero minus one.

So we've got rotation, which is our transformation, the amount of turn, which in this case is 180 degrees, so we do not need a direction.

And then about zero, negative one, and that is our centre.

C and D are now on the screen for question four.

So, question part C, a rotation 90 degrees anticlockwise about negative 2, 0.

You may have written a rotation 270 degrees clockwise about negative 2, 0.

That would land in the same position.

And then part D, a rotation 45 degrees clockwise about 3, 2, which you may have written a rotation 315 degrees anticlockwise about 3, 2.

So there is then two options on each one that isn't 180 degrees.

So, to summarise today's lesson, which was checking and securing understanding of rotation, rotation is a type of transformation.

The object and image will be congruent after a rotation has taken place, and the orientation will have changed unless a full rotation has taken place.

That was like the exception of the rule, but on the most part, the orientation will have changed.

Tracing paper is a really useful aid when performing the rotation, but also for finding the coordinates for the centre of rotation when you are writing a description, and a full description needs to state that a rotation has taken place.

You need to give the amount of turn and direction of the rotation, as well as the coordinates of the centre of rotation.

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