Lesson video

Hello, my name is Mrs Buckmire, And today I'll be teaching about combining reflections.

Now you need a pen and paper.

If you can get pencil and rubber as well, that could be helpful.

and maybe a straight edge, like a ruler, or like a side of a card or anything, that could be helpful as well.

So pause the video, and make sure you're ready.

Okay, so before we begin, make sure you pause the video when I asked you to, but also whenever you need to.

If you need more time or something, just pause the video.

Make sure you're learning at your own pace.

So even if you need to rewind the video, that's fine as well.

Let's begin.

Okay so for the trial of this card and task ahead, I want you to reflect A in x equals two, reflect B in y equals seven, C in x equals eight, and D in y equals one.

So pause the video and have a go.

Okay, so how was it? So you reflected A in x equals two.

Now all reflections are here, but I'm just going to label the images.

So x equals two is here.

So in this line, so A is one, two, three, four, away from the line.

So it's going to be one, two, three, four, away.

That coordinate.

So that vertex seems there's going to be eight that is going to be there.

The B, B in y equals seven.

So y equals seven is here.

So remember to find that out, we could just think about some coordinates, which lie on y equals seven.

So like three seven, four seven, five seven, and then plot them, and then you'll get y equals seven.

So that means B is one away from that line as one away so ended up here, so that must be B dash.

And then reflecting C in x equals eight.

So it's one away, that's x equals eight, so along that line, we imagined, if we can just think of coordinates, it'd be eight seven, eight six, eight five, eight negative two, so that's the vertical line and it's one way, one away so this is C dash, so finally this is most likely D dash.

And it is because this is y equals one, and it's one away here, so it is one away here so that point is there.

So we can see how it's being reflected.

So do pause and check your work carefully.

Especially the x equals two, x equals eight, y equals seven, y equals one, make sure you've drawn in the correct line of reflection.

So we can sometime describe the effect of combining transformations using a single transformation.

So what that means combined, means just do a multiple transformation.

So multiple rotations, multiple reflections, multiple translations.

And we can do it like sometimes, actually while I've been doing multiples, just say, oh, it's just this, okay.

And that's what we're going to be exploring.

So Carla says, "reflecting an S in M one, then M two has the same effect as a translation." Okay, let's just check this out.

So M one is here.

So it's this line.

That's M one and M two is up here.

So S is reflected in M one, yes.

I can see that to make A, and then A, reflected M two to make B.

Hmm.

So we are saying is the same as a translation.

How do we translate from S to B? Can you say it in words first? Maybe you need to do some counting? Good.

So if I'm comparing this point where this corresponding point is going to be one, two, three, four, five, six, seven, eight up.

Has it moved left or right? Not at all.

So what's that as a vector notation? Excellent.

It will be zero, and then eight, cross eight up so positive.

So I want you to explore the effect of this combination of reflections for T, U and B? So in each case, I want you to reflect it in M one, then reflect it in M two.

Okay? So pause the video, this on the worksheet as well, if you want to go to the worksheet rather than doing it from the video, that's fine.

Okay? So reflect in M one then M two, and try and write it as a translation.

Good luck with it.

Here are some of the different answers you could get.

So we already did the first one, was a translation of zero eight.

So this next one, you might want to pause and just check.

So it was two away, two away, and it was actually on the line M two.

So reflected here, straight here.

So that as a translation was zero eight.

What about you? So you actually started inside the line between M one and M two, and then it's reflected outside, so reflected below M one.

And they're reflecting to M two.

So it's currently one, two, three, four, five away from M two.

So we have one, two, three, four, five away.

So that's how we got up here.

So then we're thinking about the translation from, let's say this point to this point, and what did you get? Zero eight.

Hmm, well done.

Okay, what about the V? So V was actually on M one so you can see how it's being reflected there, they kind of overlap a bit.

And then when it's reflected to M two, it's zero eight, what, did you predict that before, when you started doing the question, maybe you're like, oh, that is going to be zero eight again.

Excellent.

That's interesting.

So it looks like that's always the case.

Let's keep exploring.

Okay, so to help you explore, I want you to do this independent task.

So these questions have been particularly chosen.

So you might notice some similarities between them, but what I want you to do is reflect S into M one, then in M two, and then describe the twin single transformation from S to the final image.

So this is on the worksheet as well, so take your time to pause.

Doesn't matter if you can't print it, you can just copy it out, but do have a go.

Maybe even try to predict before you've done it.

What you think the answer might be? Okay.

So in the first one, I'm going to do this one.

So it's one always, is going to be one always, it's going to end up like this.

That's going to be my first S dash and then reflecting it in M two, so it's going to look like this.

So that's S dash dash.

So when I translate it, it's going to be one, two, three, four to the right.

So that's positive.

And then no, I have , so four zero.

Okay.

This one looks very similar.

So what did you notice between these two questions? Good.

Both M one and M two are in the exact same position.

So both of them said M one.

What's the equation of the line for M one? Excellent.

So look at some coordinates, so three one, three two, three three.

So it must be x equals to three.

And M two? Well done, it's x equals to five.

So you can see, they both have that x equals three x equals to five.

So for this one, if I reflect it here, I get this that's S dash and reflect reflected again, that's one away from M two, so it's going to be one away, so it's going to be like this, So S dash dash.

So, do you think it's going to be the same? What did you get? One, two, three, four is the same.

So it also is four zero.

Interesting.

I wonder if you could use that to predict what the next answer would be.

So is this next question different? So now you might have already done it, but how is the question different? Gives the M one, is in the same place, so M one is still, oops, sorry.

M one is still x equals three, boy it's M two now.

It's been shifted, hasn't it? Now M two is x equals seven.

So did some of you guys think maybe the answer will be two more because it's been shifted two more? Was that true? Let's look at the answers.

Okay, so we've got the four zero, So here we have their second set of answers, and the translation eight so wasn't two more.

Hmm.

And the next one was this delayed zero.

Yes.

So even in the first case S was on the outside of M one.

So further away from M two, and in the second case S started between M one and M two, but both of them had the same translation of eight zero.

So that must be in the case where we keep M one and M two in the same place.

So the translation seems to be the same.

That's interesting.

I wonder how we could predict it.

Let's explore that.

Okay.

So what I want you you to explore it's actually come up a bit, what the translation will be.

So S is reflected in M one then in M two as shown below, describe the single transformation from S to B.

So that's what you've been doing, okay? But here, what I put is S is d distance away from M one, it might help actually, if I label M one so it's really clear.

So this is my M one, and this is my M two, and r is the distance between M one and M two, okay? So what I want you to do is describe a single transformation from S to B.

So eventually I want you to try and challenge yourself with writing it, using algebra, and maybe looking back at the examples, maybe you'll spot some patterns.

So you can pause if you feel confident.

But I think as a mathematician, I would actually specialise first, I would look up even more examples, and maybe even be a bit strategic with it, okay? Go away and have a go.

Come back if you need some support.

Okay.

So what I will do is I would change one thing and keep one thing the same.

So let's say d I change to one, and r I keep as two, three, then change to four, change to five, and I keep changing it, okay? So you might want to even make a table and say, "Oh, when d equals one.

." So this is d, and this is what my r is, then actually what's the transformation, or the translation is what we're expecting it to be.

So, for example, when d equals one, r may be, I'm going to keep having d equals one, so do it for two, three, four, and five.

And then you can change the afterwards to see what happens as well.

What has the translation be? Okay.

So I'm running out of space I'm going to make this disappear so I can draw that, okay.

So just got rid of that just so I can pull my own fresh diagram.

So here's my diagram.

So d is always going to equal one.

So that's one and here is my S.

On my S base I said it was going to be one, just to make it easier for me.

So now the first one, the r is two.

So r was the distance here between them.

So this one is going to be two.

So now when S is reflecting M one, so it was originally one away.

So it's not going to be one away, so it's going to end up here, looking like that.

And remember these are just sketches, and I know this is going to be one away, because it was one away before, and the base is one so that's why actually I know it's going to be touching my M two.

And now reflecting M two.

It's going to reflect like that.

So that's my S dash dash.

So now what is the transformation from S to S dash dash? Let's see.

So I moved, this point moves one, two, three, four.

So if it's moved four to the right, What's that as a translation? Excellent, four zero.

And then you can have a go and try and imagine if that length instead changed to a three.

So now that means that actually M two has moved up.

So M two has moved up to here by one.

And so actually, how does that change? Do I make a new diagram, okay? Let me give you some more support.

Let me show you on.

But if you feel confident, have a go, but also I want to explore using GeoGebra as well.

Okay, so I've set up the situation on GeoGebra so GeoGebra is a fantastic website, and it's a free website, and I must master, you made this up look bit like this situation here for me.

So what I've done is S is a one, now what was that called? That was our d, wasn't it? So here, this distance of d which at first I was keeping constant, which was one it's out there.

And now what was this called? Was it r? Yeah? So our r is there was two, as you wanted.

And as we expected B is here.

So we're mapping, what happens when S goes to A and then goes to B.

Let's see if I can use this pen.

So at the moment S has been reflected in M one.

This is M one.

Sure I can note here, M one, this one is M two, reflecting M one to get to A, and that's reflected in M one and two even to get to B.

Okay, so as we expected, we can see that at the moment, it's being translated one, two, three, four, okay.

So I'm going to give you some more resources If you're struggling to do it yourself, and I'm going to make r one more bigger.

So if I make it one bigger to here, what is the translation now? Pause video, if you need to, and write it down, okay.

If I move it to here, what is the translation now? I'm finding if I make r five now, what will it be? And now fine let's do one more.

So that's five big for r.

And this one's six big for r.

So what's really important is that you're collecting this data and writing it down in the table, okay? So that was six, five, four, three for r, and two for r.

Okay, so what about if actually was changed d? So let's make d two now.

That's interesting.

And B just swapped.

Hmm.

So let's keep r as two constantly, and now S is two, sorry, d is now two as well.

So this in your table is going to be where d equals two, and r equals two.

And then d equals three, and r equals two.

And then let's do d equals four, and r equals two, and let's see what happens.

So we're constantly seeing the translation from S to B, that's the translation you need to write in each one.

Okay.

So that was when d was two, and d equals three d equals.

Sorry, so this is d equals three.

And there's d equals four.

It almost comes off the grid.

Okay.

So what is the translation from S to B? Okay.

I've been amassed enough data, hopefully using all that information.

you can start to think about some rules.

Pause the video, and just write down what you notice.

Okay, I know it's a further item, I just want to continue with my working out what I was doing.

So when d equals one and r equals two, we got four zero.

And when d equals one and r equals three, we got six zero.

When d equals one and r equals four, we got eight zero.

And when d equals one and r equals five, even if you didn't do it, what do you think it's going to be? Well done, it was 10 zero.

Okay, what about when d was, I kept d the same, no sorry, yes.

I kept r the same even.

So I kept r the same.

So I kept r two, and I changed d, didn't I? So I changed d to two, and I changed d to three using the GeoGebra.

So when d was two, and r was two, we got four, zero.

And then when d was three and r was two, did you do it? We got four zero as well.

That's strange.

It looks like four zeros going on.

So we have four zero here with this line, and we have four zero here as well and up here.

So what do all of those have in common? Interesting.

They all have our r value.

Excellent, yeah, at two.

Okay, any others? Interesting things.

Yeah, we had six zero when r was three, eight zero when r was four.

And I asked you to predict the next one.

A lot of you said 10 zero.

And that was very surprised.

So what's that relationship? Oh, good.

It's always double the r value.

How could we write that? Excellent.

The translation vector from S to B is two r zero.

Really, really well done if you've got that.

Now if you would like to explore some more, you could explore what happens if you did M two first then M one maybe.

That's for you to do.

Maybe you can already predict it, but I'm going to go through why that happens.

If you want to understand it, list this bit.

So why, I always used to ask why does that happen? Like why this pattern exists? And I think this is something you can understand, okay? So, if you explored it, you would have seen, I'm going to take this distance to be the base, and I'm going to call it b for base for the base of the triangle.

Okay, so all these are b long, which could be helpful.

Okay.

So what is the distance between here, from this point to this point? Yes, it is d, because we know of reflection.

Each corresponding vertex is equidistant from the line, as in, it's an equal distance on either side.

So if that's d, then the other side is going to be d as well.

Okay, so that's fine.

And then we have this distance now.

I don't actually know what this distance is, but I do know that is going to be equal to this distance, correct? So let me just give me a letter, any letter, c, I'll use c.

So they're both going to the same.

I don't know what they are, but they're both going to be the same.

Okay.

So now if I'm translating this point, if you let me highlight it, yes, to this point, what is it going to be? Well, it's going to be d, plus r, plus c, plus b.

So the movement, so to the right, I'm going to write in words first is going to be, it was d, plus r, plus c, plus b.

Yeah? So, this d plus all the way across which is r, you could do, plus d, plus b, plus c, but I think it's easier just to all as r.

And then plus c, plus b, hmm.

That still doesn't come up as two r, or does it? We said d, plus b, plus c is the same as r, isn't it? So d, plus b, plus c, is the same as r, so round them all and I'm going to write r, So now I have r plus r, which equals two r to the right.

And that as a translation is two r zero, yeah.

It Works out.

Don't worry if you didn't understand that.

Okay.

That is really applying the algebra.

But I think maybe if you look back at it, maybe you could understand that, okay? But well done if you do.

Excellent, excellent work there everybody.

If you had a go at reflecting it twice, and had a go at translating, trying to describe that translation as a vector, then I think you've done a fabulous job today, and you should be really, really proud of yourself.

Okay.

I would love for you to do the exit quiz because that will really help cement your understanding and check what you have learned today.

Have a lovely, lovely day and thank you for all your hard work.

Bye.