Lesson video

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Hello, my name is Mrs Buckmire.

Today, I'll be teaching you about combining translations and reflections.

So first make sure you have a pen and paper.

If you have a pencil and rubber and ruler or straight edge, that could be helpful as well.

So pause and make sure you're ready to begin.


Remember, you can pause the video whenever you like.

Do pause when I ask you to, but also whenever you need to.

And rewind and listen to bits again if you don't quite understand.

That can sometimes help.

Okay, let's see what the trial this is.

So before I try this, I want you to describe the transformation from A to D, D to C, and C to A.

And C to A is probably the trickiest one.

It might help if you had tracing paper, but if not, I feel like maybe you can still figure a bit out.

And then, use the information that's been given to us, that A to B is a translation by four negative nine.

Can you find a missing coordinate of all of the rectangles? Okay? So this is quite a challenge, but I think it would be really useful to kind of get used to the position grid, different coordinates, and a bit of working.

So do spend five to 10 minutes on this.

Have a go.


So from A to D, what did you get? Good.

It was a translation.

Did you write in vector notation? Excellent.

So I would go from negative six three co-ordinate, to four one.

So, how far do you have to go to the right? Excellent.

And then how far down? Good.

So it should be 10 negative two.

What about D to C? Good.

It was reflection, in the line, in the x axis, yes.

So you could do x axis or y equals zero.

They're the same thing.

How about C to A? That was a tricky one.

Rotation, yeah.

Rotations are hard to spot the centre.

What was the centre of rotation? Excellent.

It was negative one one, and it was a 180 degrees.

If you got rotation, how many degrees, then fantastic.

The centre was negative one one.

Excellent, if you've got that as well.

So then all the coordinates, so how did you work out the coordinates first? Good.

So what I would do is see, actually, what is the length of the rectangle, and I will use this, What's this length? So that's the difference in the y values.

So that distance is four.

So, it's going to be ending in seven.

I know that it's on the same line as this other one.

And the exact value is going to be the same.

And then here you can work out this one, because it is negative nine in the x value.

And the y value is going to be the same as this y value.

And then kind of using the reason that we know now that this side is four, and we know that this side is three.

We can then work out all the other ones as well.

That's what you should get.

Pause the video and check your answers.


So for the connect, what I want you to do to describe the transformation or combination of transformations between each pair of triangles.

So from S to A, S to T, S to B, B to A, B to T, B to S like that.

Do spend five to six minutes to see how many you can get in that time period.


Just gave myself some room to write.

So there are lots you could get.

So let's say from, let's do the axis.

So from S to A, what did you get? It is a reflection.

And, below this line, so imagine points in the line, so we have four one, four two, four three, x equals four.


What about S to B? Excellent, it's a translation.

So from this to this corresponding point is one two, three downwards, so a translation in the vector zero negative three.

One reason to remember it, it is negative because it's going down.

And then S to T.

No, it's not a rotation.

So some people might have thought rotation about this point, but then actually S would end up looking like, like that.

Like with the right angle up here.

So that is so bad.

Wait a second, sorry, Mrs Buckmire how messy are you? Right.

It would look like, from this point to this point, to this point right here, it would look like that if it's rotation.

So that's not correct.

So this one, we can use a combination.

So what combination could we use? Excellent.

So could go to S A, so it could go S to A, and then A to T.

Or even S to B, and then B to T.

So in that case, it would be a reflection, in x equals four then, I don't know why I used capital it should just be small.

But then a translation of zero negative three.


So hold that one in your head from S to T, and let's see what Anthony has to say.


Anthony says the order you reflect and translate doesn't matter.


So keep what we just did in mind, where we had reflection x equals four, and then translation in, what was it again? Good, zero negative three.

So this was to get from S to T.

Explain why the student is correct in the case of mapping S to T.

Is this statement true for all translations? So in our case would be reflection then translation.

If you did translation first, then reflection, would you still get to T so it says we do, but do you trust them? And it this statement true for all translations, okay.

Just pause and have a think.


So, first if we did the translation here first, so we went down three first, so one, two, three, ah, we'd get to B.

And then we did the reflection.

We would end up at T, so actually, if we swap these round, we do, it does work out the order doesn't matter, but will that always be the case? Hmm.

Did you come up with some examples maybe in some of your own questions where it didn't work, maybe where you reflected first, and then you translated.


We'll look more into that later.


So for your independent tasks, I do actually want you to look more into it.

Now I'll give you an example, and this is Yasmine.

So Yasmine is a student exploring the effect of combining a translation and a reflection.

And actually she's used a diagonal line reflection here.

So just this line is her line of reflection.

And what we can see is that, first you reflect in the dotted line, and then translate by the vector of negative two two.

So we can see that, for this one.

So she did from A to B one first, so the translation first, and this one, she did translation first.

And then she reflected it and she got to C.

So let me explain.

Where's my.

Yeah, so here, she did A to B one, and that was by a translation.

And then from B one reflected it, so we can see it's one, two, three away, so one, two, three.

So that's how we got to C.

So A to B one to C one so then it was a reflection.

So then you'll just reflect it first, so a reflected one, two, three, so reflected one, two, three.

So I got to here, and then she did the translation of negative two two.

So negative two two.

And she ended up at C.

So, A to B two, to C, was a reflection first, and then a translation, okay? So what I'm going to do is I'm going to give you some reflection, a reflection line, a line reflection, and a translation.

I want you to do it in different orders, okay? Hopefully that makes sense.


So here, use the line of symmetry shown to compare the effect of reflecting then translating, and then of translating then reflecting, for each of the vectors, one zero, one two, and one one, with the same line of symmetry.


So, here it is, you can do it on separate diagrams. If you like, you can just do it all on the same diagram.

So this is for one zero.

This is on a worksheet, for worksheets if you want to use it off the video, this is for one one, this is for one two, And that is you've finished.

So do make sure you pause there, look at the worksheet and do how I've done it.


So question one.

So reflecting and then translating, so let's see which answer's which.

So here, this green one, let me do it in green so it's the same.

This is it having been translated right now.

So we're translating it to one zero.

So that's one to the line, makes sense.

And then it's being reflected, so it ends up here, right.

So, the red one, this is the reflection, ends up here.

And then when it is, so this is the final one, let me label the final one A there, to make it easier.

And then when it's being translated, it translates to here, so it's not the same.

So here, the order did matter.




What about here? So here, for one one, when it was done first, which is A, you end up here, and then the final one, ended up being here when it was reflected.

So that was the final.

So, and then what about the other one? So it was reflected first, it ends up here, and then it was moved one one.

So that's one right, one up.

Oh, it ends up in the same place.

So it did work for this one.

Didn't work for the first one, the order didn't match up, in the second one, the order didn't matter.

That's strange.

I wonder why.

Hmm, do you notice anything between one zero and one one?> I wonder.

Compared to the line of symmetry maybe.

Okay, I just want you to show you this one.

So if it's reflected first, it'll be there, and then it's translated one two, that's one to the right and two up.

So it moves to here.

So that'll be your final one.

Or if it translated first, I moved one and then two up.

So it's like up here off the grid, and then it is reflected.

So it'll be here off the grid, but the other one was not off the grid.

So it doesn't work.

One two didn't work either.

So only one one worked, with that diagonal line.


So the shape undergoes translation or reflection.

When do your following combinations have the same effect? So we've seen lots of cases where it has worked and somewhere it hasn't worked.

So why is that? What I would like you to do is look back at those examples and maybe even create some of your own examples, and just think, okay, what do I notice? What's the relationship here? So for, do they have the same relationship? Does it have the same effect when we do a translation then the reflection, or the reflection then the translation.

Just have a look through our notes or look back at certain examples, you could scroll through.

Pause the video and just think about it.


So, hopefully you had a chance to have a look at those different examples.

So I'm going to show you this example.

So this was one that did work, and this was from independent task, and it was when we had that diagonal one, the one we had one one, and this was also one that worked.

So here it worked when we had the translation between zero negative three and this vertical line, which is x equals four.

Someone told me once, and when I did this, but maybe it has something to do with the line of reflection and whether the vector's parallel.

What do you mean by parallel? So parallel, they're lines that never meet, but let's say so here's zero negative three, if I did that as an arrow, that would just be a downward arrow.

From here to here.

What would one one look like? So we'd go one across, and one up.

So it'd be from here to here, to look like that.

So what do we notice about the arrows, and the lines of symmetry considering what I just said to be fair? There are parallel.

So parallel means they never meet.

They're parallel to each other.

I wonder if this is always the case.

Maybe you could have another go at this activity, using two two, cause that would be parallel, going two across then two up.

Or maybe this activity using zero two, and then x four swapping those rounds.

So pause, if you want to check out that hypothesis, that actually, if the vector is parallel to the line of reflection, and the order doesn't matter.


Did you have a go? It does work out, doesn't it? Interesting.

So if I was going to actually write it as an official mathematical statement, what I would say is, the combined transformation equivalent to a translation and a reflection is commutative, when the translation vector is parallel to the line of reflection, so many words, but do you know what? You understand and know all of them.

Commutative, you've learned that before.

So A plus B equals B plus A? Is there any operation where the order doesn't matter? And then parallel must be in the same direction so it wouldn't meet.

If it was a line, it wouldn't meet.

So parallel to the line of reflection, which, you know, the translation vector.


You've understood that.


Now, even if you didn't, I think you've worked really hard today, and it's really good.

Even if you just had a go at translating and reflecting, you just practise that.

That's really, really great.

So really, really well done.

Hopefully you got to check over your work.

And, I would love it for you to do the exit quiz because that will help with your learning.

And I'd just like to say thank you for participating in the lesson today.

Have a lovely day.