Lesson video

Hello, my name is Mrs. Buckmire, and today I'll be teaching you transformations: rotations.

Now first, make sure you have a pen and paper.

There will be a plat screen in the try this where if you could ask your parent or carer to use some scissors, if you have paper you can cut up, and a coin, or like a rubble sharpener and that could be well, that would be very, very useful.

So pause the video and get those things, if you would like to get involved in that practical.

Okay, before we begin, when will I ask you to pause? When should you pause? When should you pause the video? Good.

Whenever I ask you to, and whenever you need to, okay.

Remember, you can go at your own pace, rewind in the video if you need to.

listen to things again, that can be helpful as well.

Okay.

Let's begin.

Okay, so if you would like to get involved in the practical, make sure you have this equipment.

You have scissors, or you asked for parental or carer permission to use them.

You've got coin or sharper rubber, and you've got paper that you're allowed to cut up.

Okay, it doesn't really matter about the sizing.

It doesn't have to be that big, but just any scrap paper or anything.

Okay, so we have the paper, all I want you to do is cut a right angle triangle.

So like that.

Now I'm going to use this one that I prepared earlier because it's colourful, so it can be seen easily by everyone, hopefully.

Okay, so what I've done I've actually labelled the front F and put a right angle in at the right angle, and then kind of twice labelled the upper angles as well.

So I'm going to label that one with three and this one's got two.

Okay, so what I want you to do is place your coin or your object anywhere on the page.

Let's just put it in the middle for me here and then place the triangle anywhere.

I'm going to place it here.

And now what your challenge is, well first let's just trace it before I tell you about the challenge.

Trace it like this, whoops that was not very well traced.

Okay, maybe you've got a ruler or straight edge and you can make it even neater.

So I'm going to label this A, this point B, and this point C, those corners even.

Okay, So your challenge is try to place this triangle in another position on your piece of paper, where the distance of each corner to the centre of the coin is still the same.

And the front is still facing forward.

So what I mean by that is this distance from here to the centre, which we're going to call A, and this distance, or C, to the centre and, or B to centre that kind of distance there always remains the same, but you move its position.

So like A is closest? Yes.

B seems like maybe it's next closest.

And these are how, if you have a piece of paper that you can cut, you can actually also say, if you have a ruler then that's fantastic, that's way easier.

But otherwise, even if you don't, I just don't want you to be put off.

You can actually just put the paper that you're allowed to cut.

Just kind of map on.

So like A to the centre, so the centre is like my hop here.

So to A, that's the edge.

So that's A, and then from B to the centre is around there.

So label that B and then the other side we can do for C.

So what we can see is actually, so C and B here are actually around the same.

So C and B needs to be around the same distance away and A needs to be closest.

So where is it that A is closest and B and C are around the same for me, but you ought to be a bit different.

So you just have a little play of where will that work out.

You can use a ruler, I'll be using this, if you have a ruler, but otherwise just try and figure out what positions or is its look.

F must be forward You can't just kind of turn it around.

This wouldn't be correct because B and C aren't the same distance, but you can't just turn it around.

I always want the front facing forward.

Maybe how can you turn it to make it the same everywhere? Just pause the video and have a little play.

Okay.

No worries if you don't have pen and paper, you can still do the "Try this" and I've made this worksheet for you.

So here you can see you that the triangle has been placed on the unit grid.

And first I want you to tell me what is the distance of each of the triangles vertices to the cross.

Then I want you to draw the different positions that the triangle can be moved to, so that each vertex remains the same distance from the cross and the front face, the shape labelled F, remains seen.

So it's the same problem, really.

It's just that it's been done on a piece of paper, which is more than fine.

Okay, pause the video and have a go now.

Okay, so let's look at this then.

So I'm going to do the practical first and then I'll have a little explore with the other one.

I'm going to have the triangles actually have drawn slightly different, so here it looks like maybe it would work out.

So A, I want to be the same distance and then B and C around the same distance from centre.

Maybe if it was like here.

Yeah, that would work out B.

Yeah, that works out for B so here works out for me.

Do you get more than two? So maybe this position to A needs to be closer to B and C about equal distance.

This looks about right for me.

So even if you can't measure it, it's fine for you to just predict it and just have a little play.

What about here? No, A's not closest there.

So that doesn't work out.

Maybe here.

I want B and C to be the same.

So maybe there? Let's see.

So feel free to check it if you want to make sure you're right.

C, B, oh, no, that was meant to be B.

Yeah B and C.

Yeah that works out so I'll cut them around here, but do you get more than four? So A closest and B and C equal distance.

Could that work out here? That looks like it might, let's see.

C needs to be a little bit closer there, B is right.

Oh, this one works.

Okay.

I wonder how many there are.

How many do you think there would be? Right? Let's use GeoGebra and try it out.

It's the diagram from the Try This and now what different positions.

So what I can actually see is, so we had the position up here We had one here and we had one here.

Now what you might notice, I'm using GeoGebra, this is a free online app.

It's really awesome that maybe when I'm turning you can see actually are little places as well.

So actually this would also work out.

This would work out, here would workout.

So how many different triangles satisfy the conditions I gave you? A hundred? A thousand? It's actually infinite.

So I can only move this by a small amount, but actually I could move, it by even smaller.

You just move it by a small, little, little fraction of an amount each time you can have an infinite number.

Now, what I am actually really interested in is actually what is going on to this point.

So how does this point change this one closer to the centre? So the moment I can see, oh, it's two away.

So it's two horizontally away or two other units square.

So it doesn't even matter about being horizontal.

So, another place that point could be we saw was actually here.

Another place was here, and another place was here.

Let's plot some more places it could be.

Sorry, just checking that those were the right points.

Yeah.

Can you see? That's where it's each time.

Okay, so what about, let's plot this one here.

Okay, Let's plot another one.

Should put one here and I want to know, maybe you can start even predicting it, What shape does it make? So all of these plots, if I keep plotting it, what shape will it create? I'll plot a few more.

How many more should I plot? Two? Okay.

Two more.

Let's plot one here.

Whoops.

I moved it while I was plotting it.

Here we go.

Plot.

And one more, where should I do? Which side? There? Okay.

So what shape is being created? Hopefully you can see it X.

I mean, it's a circle.

That point creates a circle.

And every point on that circle is two units.

So equivalent to two of those horizontals away from the centre.

Okay.

And that this point, if I plotted them all, how do you think you can even compare it to the red one? What is it creating? Good, it's also create in a circle.

So what we can notice is that each vertex creates a circle from the centre.

And so what do you think will happen? And if you have a practical, you can have a go.

What do you think will happen if I move this centre, if I moved the coin, what would have happened? So let's say if I moved the coin downwards, what do you think is going to happen? Let's see, the circles all move as well.

What about if I move it further away from the triangle? Yeah.

Would the circle get bigger or smaller do you think? Good.

Some of them will get bigger.

So this one in the smallest circle got bigger.

Well, actually this one, yes, it was bigger, but really it was just, it was, it has still has the same distance from this cycle.

That length doesn't change, but yeah, all the circles got bigger.

What about if I change the shape? Here we go, if I just changed one point only one circle changes.

So, you can have a little play with your practical as in, if you move the point in a different place, where could you now draw it and how it changes and, yeah, just have a, have a little thing.

Just imagine what would kind of happen if different things occurred.

So that's why GeoGebra is really cool because you can actually see it play out, but also through the practical it's nice to imagine it as well.

Okay.

I hope you enjoyed exploring a bit.

So I just wanted to kind of define what rotation is.

So rotation is a type of transformation, what you would have seen is we transform, we moved that shape, the triangle, over outward to over in our space.

So it is a type of transformation, a rotation tears objects.

You would have seen actually with GeoGebra, that the object turned and the size and shape stayed exactly the same.

So it was always a triangle and the size of the triangle never changed, but the orientation changed.

So kind of the direction it was kind of pointing.

To complete or describe rotation, we need to know.

So what information did we need to create those images? Well, we had different angles and actually if I want to be very descriptive and very precise, actually even about where it is, I'd need to know the exact angles, the direct rotation, if it clockwise or anticlockwise.

So remember clockwise is the way the clock goes like this and anticlockwise is the opposite direction.

Should you draw it like that, really.

By that, anticlockwise.

And then we also need to go know the centre rotations, So here I had it as a point because that was my centre rotation on a coordinate grid to be given coordinates where it is.

So pause the video, and add this to your notes.

Let's go through an example.

Rotation shapes.

The octagon P is rotated about the origin.

Hmm.

Where is the origin? It's at the centre.

So it's at That's the origin.

You can write that in if it helps.

And I put a plus there to help you as well.

So A is the image after 90 degree rotation clockwise about the origin.

Hmm.

Where do we think it will end up then? Where will P be rotated to? So it's 90 degrees, clockwise or clockwise left to right.

If I had trace in paper, I would actually trace around the centre, trace around the shape.

So I'd put my trays and paper.

So it looked like this.

So that whole area is where I would trace around this plot and this part.

And also actually drawing a little arrow going upwards like that.

So now where am I expecting that arrow to point after a 90 degree rotation clockwise? Good.

It's no longer, I mean, north it's going to be east.

So then I would have to turn it so that it went out east.

This doesn't turn with me, but it would now face east and certainly not expect it to be there.

Is that where you imagined it was going to be? Excellent.

And now I would carefully twice around it kind of think about key coordinates, like key corner.

So that could help.

And then I like much more up.

So here I'm able to trace around it like this.

On yours that's how you'd work it out so it would be there.

So that would be a 90 degree clockwise rotation.

So what about if we had B as the image after 180 rotation clockwise by the origin, have a little go I'm imagining where you think it would be.

If you have a tracing paper, you can use that as well, and have a go.

Okay.

So did you pause it and have a go? So after 180 degree rotation, so again, I'm going to actually use my tracing paper outline.

I'm going to turn it 180 degrees now.

So if my arrow was originally pointed upwards, where do I want my arrow to point? Good, downwards.

So at the moment, it's upwards at the moment, it's like in line, in the direction of P, really.

I'm going to want P to be upside down.

So let's turn until P is not upside down.

So that was A, sorry until P is upside down.

Ah, there, that would make me upside down wouldn't it be going all the way around.

So here is where it would be.

Is that what you imagined it? Nice.

Okay, Let's just check your understanding.

So the oxygen Q is rotated about at 180 degrees.

Where will the bold black point be on the rotators shape? Pause the video and have a go, Okay.

There are different ways you can do this.

So first let's say you don't have a tracer paper, so first we know from that GeoGebra activity, but especially from looking at that, that actually the distance needs to be the same from the centre.

So actually, if I consider that it's like one across one to the left and two up.

So like it's this kind of, diagonal of A one by two rectangle then actually in these black distance.

So it could be D, it could be C or it could be B.

So it's definitely not A.

Now for my imagination, for my mathematical sense, I'm thinking 180 degrees is going to be actually within our bottom quadrant is not going to be D because its going to have turned too much.

So it's not D, so we know it's not D and it's not A, so is it B or C? If I'm imagining it turning, I think B will be too far.

So I think actually it's going to be C and we can kind of see it sort of symmetry within how the lines are opposite each other.

So I think it's going to be C.

If you have tracing paper, that would be the best thing to check.

Let me show you how.

So, what I'm going to do is if I had traced them, I would trace, around this and let's see when I turn it.

So any turn 180 degrees there, here I'm turning 90.

And I remember Q is facing like how we normally read it.

So I want it to be upside down.

Don't I? So that would be 90.

I, yes, a D was a 90 degree turn clockwise.

So that's not right? Yes.

It does seem to be C.

There we go.

That's 180 degrees, turned about.

Well done if you've got that right.

Okay, let's have a little check also about describing rotations.

So how many ways can you describe a single transformation from hexagon S to hexagon T? What information is needed to describe a transformation that is rotation? Good.

So we need to know the angle of rotation, the direction of rotation, and the centre of rotation.

Pause the video and have a go.

Okay.

So what did you get? Was it 90 degrees, clockwise or anti-clockwise? Clockwise? About four, six.

Excellent.

Was that the only answer? Good.

There was another one.

What was the other one? Fantastic.

Could be a 270 degree turn.

Anti-clockwise about four sets.

Well done if you've got both of them, if you didn't, try and see why, why do they both work? And maybe even have a little practise of the one that you didn't get.

Okay, you are ready for your independent task.

There are three questions.

Now question one is to generate five statements describing the angle and direction of rotation between two shapes.

And there's an example for you.

So there's no centres here.

It's not set on any position in a space, but just having a little go at that.

But two, I want you to describe the following transformation, so from A to B, A to D, A to C, and B to C and for C for three, even there's a vertex there is marked is it should be.

A rotation 90 degrees clockwise about , where will the Vertex end up? and for B and the rotation 90 degrees anti-clockwise about see a rotation, 180 degrees about point and the rotation 90 degrees clockwise about.

So pause the video and have a go at these.

Okay.

So make sure you've done one, two and three.

So where's your one.

Now there are loads of statements.

So I'm just going to give you a few.

So you could say A to D is a rotation of 90 degrees anti-clockwise, you could say B to E is a rotation of 180 degrees.

Wait, why does A to B? have always anticlockwise and B to E only has 180 degrees, like no anti-clockwise clockwise? Good.

It doesn't matter.

So B to E 180 degrees we'll go clockwise or anti-clockwise we'll still end up from B getting to E.

Does the order matter there? So does it matter if I say B to E or E to B? No so not in that case but in every other case it does.

So you can't say, "Oh, D to a is rotation of 90 degrees anticlockwise" because that wouldn't be true.

So make sure you check your answers really carefully and that issue, the order does matter there sometimes.

So E to F is a rotation of 90 degrees clockwise.

Now there are lots of different statements, so well done if you've got all five.

Okay, for the independent task question two, describe the following transformations.

What did you get for A to B? Good it's definitely rotation and it should be 90 degrees clockwise about A to D? You should have got rotation of 180 degrees about.

A to C? Rotation, 90 degrees anticlockwise about and B to C rotated 180 degrees about.

Well done if you got those.

Pause and check your answers carefully and maybe anything you've got wrong, have a little play and check, Oh, is this correct? Okay.

These ones are quite tricky, especially if you don't have tracing paper, but you can definitely have a go at, imagine it.

And having like, where do you consider it would be, and then trying it.

Okay, this final one.

So it should be , , , and.

Again, have a play and just had an idea.

If you don't have tracing paper, you could actually just use normal paper.

Like this stuff is normally thin enough that you can actually press hard enough to see it or just draw with a thick pen or maybe go to a window and see it.

So if any of these, you've got wrong, go and check out.

If you don't have tracing paper, be creative okay, I really want you guys to leave to later do this.

Okay, and that relates to the Explore.

So all of you guys now have some ideas of ways you can work these out.

So what I want you to do is describe rotation that would create a square from the given shapes.

Mark, And my thing is, to mark the centre rotation with crosses and label them.

Okay.

So what I would say, is first think about how you want it to fit on the square.

So maybe even draw it inside the A image B image, C image, and D image and then decide where the centre needs to be and label them.

This is a challenge.

So do you have a little play and if you need to come back to support, more than fine, just return to this part of the video.

Okay, So a bit of support.

So what I've done is I've actually drawn it inside for you.

And then I've also even actually created the points, the centre of rotations.

So one reflects rotates.

One is a centre rotation for A, two of the centre rotation for B, C has centre rotation three, and D has centre rotation four.

So I've labelled it clearly, hopefully enough for you.

And then now you need to try and describe it.

So if you feel confident, feel free to pause.

There's always a bit more support as in, for each description, you can fill in these gaps.

If you fill in these gaps and decide if it's clockwise or anti-clockwise, depending on what your degrees are, then that could help you as well.

For everyone now makes you pause and have a go.

Okay.

So where you could have got is shape a, has been retained about point 1, 90 degrees anti-clockwise.

Now remember there are different answers.

This is just the one that follows on from the support and the way that I have done it.

Shape B has been rotated about point 2 at 90 degrees clockwise.

So what could be another answer for that? Good.

It could be 0.

2, 270 degrees anticlockwise.

Point C has been retired point 3, 180 degrees I've showed them clockwise, it doesn't really matter here.

And point D has been rotated about point 4, 180 degrees clockwise.

Really, really well done, if you got that or anything that worked, that worked out fantastic.

Well done today, everyone.

So I know rotation can be a tricky topic, but hopefully you've had a go and really I wanted you to do to be thinking about is just how can you imagine it turning and where do you expect it to be? I hope you've enjoyed today's lesson.

Why would like you to do is take a minute to write down three things you've learned.

When you've done that, make sure you do the quiz.

The quiz is really helpful and I've given you some feedback as well.

So it means that you can see what you've got, right and the things you've got wrong.

You can look at the feedback and hopefully that will help you.

Have a lovely day.

Bye.