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Hello everyone, it's Mr. Millar here.

In this lesson, we're going to be looking at 3D coordinates.

So first of all, I hope that you are all doing well, and just in case you haven't watched any of my other videos online before, let me just briefly introduce myself.

My name is Mr. Millar, I am a math teacher at a secondary school in central London and I am doing a series of two lessons for you.

So the first one, this one, we are looking at 3D coordinates and the one after this, we are looking at a painted cube.

They're both going to be lessons about 3D shapes and they're also going to be pretty challenging lessons so make sure that you are ready for that, make sure that you have your resilient hat on because these will be tricky but also hopefully some really interesting problems. So let's get started with the try this task.

As you can see, we have got a 3 by 3 by 3 cube and each of the dots that you can see on the cube has an X-coordinate, a Y-coordinate, and a Z-coordinate.

Now, you've probably all seen before, I'm sure you've all seen before, X and Y-coordinates, so you know that X-coordinates go across and Y-coordinates go up.

But in this 3D cube, we have a Z-coordinate as well and you can see from the diagram underneath the cube that the Z-coordinate goes into the page.

So let's have a look at our first example.

If we have a look at the red dot, we can see that the coordinates are one, three, one where the first one is the X-coordinate, the three is the Y-coordinate, and the second one is the Z-coordinate.

So first of all, if we look at the X-coordinate, we can see that the X-axis is going this way so anything along this line here, for example, would be an X-coordinate of one.

And we can imagine that the X-coordinate is also one for all of those points up there.

The X-coordinate would be two for any of these blue points here so X-coordinate of two would be here and up here as well, so yeah.

That is the X-coordinates.

The Y-coordinates, I've just brought these up quickly, the Y-coordinate tells us how far up we are.

We can see that if we were on this first level here, we've got a Y-coordinate of one.

Second level is a Y-coordinate of two and this third level, anywhere on the top of this cube is going to be three.

And we can see that this red dot is on that top level where the Y-coordinate is three.

Finally, the Z-coordinate, well, we can see that the Z-coordinate looks at how far into the page we are.

So we can see that at this red line here, we are one unit inside the page so the Z-coordinate goes that way.

So we can see that the Z-coordinate is one.

Okay, so that is how this works.

What I've got now is three more coordinates for you to try.

So here they come, you've got a yellow coordinate, a green coordinate, and a purple coordinate.

So for each of these, you need to give the X, Y, and Z-coordinates for each of these three coordinates.

So pause the video, four or five minutes, try to work out these three coordinates.

Pause the video now.

Okay great, so let's go through these nice and quickly.

So first of all the yellow coordinate, well, the X-coordinate is going to be one, the Y-coordinate is two, and the Z-coordinate is zero because we're not going into the page at all, we're kind of still on the front face.

The green coordinate, the X is two, the Y is three, and the Z is two because we're two into the page.

Finally, the purple coordinate is going to have X-coordinate of three, it's going to have a Y-coordinate of one and a Z-coordinate of two as well.

So, those are the three coordinates and on the next slide we're going to have a look at this in more detail.

So when you're ready, let's move on to the connect task.

Okay great, so here we've got the same 3D cube with three coordinates, X, Y, and Z, and this time what I want you to do is I want you to copy the cube as best as you can and then write down on the cube that you have written in each of these three coordinates.

So for example, zero, three, zero.

I know that the first one is X, and then Y, and then Z.

So I'm looking for an X-coordinate of zero, so I know it's going to be one of these coordinates up here.

The Y-coordinate is three and the Z-coordinate is zero and I can see that this point here is going to be my first coordinate, zero, three, zero.

What you need to do is pause the video now to copy down the cube.

And on the cube, mark on the three other points, and also down here below, write the coordinates of a point that you can't see on the grid.

So a coordinate that is hidden within the cube.

Pause the video now to do this, four or five minutes.

Okay great, so let's go through these now.

So the first one, one, one, zero.

Well, that is going to be here.

The X-coordinate is one, the Y-coordinate is one, the Z-coordinate is zero.

The next one we'll do in green, three, zero, zero.

So we want this one here, and also, it's worth mentioning that this one here is on the corner, the X is three, Y is zero, Z is zero.

And the final one, we will use black for the final one, three, three, two.

So our X is three, Y is three, and Z is two.

Well, that is going to be this one here.

Okay, and the final question, write down the coordinates of a point that you can't see on the grid.

So have a think, if you haven't already, of a coordinate that you wouldn't be able to see on this cube.

Okay, well there's actually a lot that you could've thought of.

One idea is having a think about the point one, one, one.

Because if you have a look at the purple point, which is one, one, zero.

If you imagine the point one, one, one, well how is that going to be different, is that it's going to be a little bit inside the cube like that because it's going to be one in the Z direction which means that we won't be able to see it on the outside of the cube in the view that we currently have.

Okay, we're now going to move on to the independent task where we're going to keep the same cube but the task is going to look very different.

So when you're ready, let's move on.

Okay, so somewhere in the grid a piece of treasure is hidden.

So we are still looking at the same grid, and you are going to get a series of clues which will lead to the treasure.

So the first clue is here, the X-coordinate plus the Z-coordinate equals Y.

So we can write X plus Z equals Y.

Now on the right-hand side, I have given for you all possible 64 coordinates where the treasure is hidden.

So it's going to be in one of these coordinates and we're going to need to work out which.

The first clue is that the X plus the Z must equal to Y.

So what I'm going to do is I'm going to figure out which of these 64 coordinates it could be.

So for example, if I look at the first one, zero, zero, zero, so I know that my X is zero, my Z is zero, and my Y is also zero, so does this satisfy X plus Z equals Y? Sorry that should be an equal sign.

Yes, it does.

Because zero plus zero does equal zero, so I could circle this first one, you might want to write down zero, zero, zero, because the treasure could be in that point.

If I look at the next one, zero, zero, and one, well that is zero plus one equals zero, and that is not true, so I can cross off this second coordinate here because X plus Z does not equal Y.

And I can keep on going, so I know that this one doesn't work, this one doesn't work, this one doesn't work as well, but when I get to my next one, zero, one, one, well, X plus Z is going to be zero plus one equals Y, which is one, and yes, I can have this coordinate because zero plus one does equal one.

So, what I want you to do is I want you to go through all of these coordinates and see which ones you end up with before we move on to the next slide.

So pause the video now and go through all of these coordinates to see which ones, X plus Z equals Y.

Pause the video now.

Okay, brilliant.

So on the next slide, I will show you the coordinates that you should have left.

Okay, so on the right-hand side I have put in a box the coordinates where X plus Z equals Y.

So those are the ones that you should have had, how many have you got? We've got 10 possibilities now.

So here's the final slide, the explore slide, and there are five more clues, clues two through six, that will get us the treasure eventually.

So here are the five remaining clues.

We can go through the first one together so you see how this works.

So the treasure is not to be found at any of the corners.

And if you think back to that three dimensional cube, you know, any of the points on a corner will have the X and the Y and the Z-coordinate either be zero or three.

So we can cross off a few coordinates here.

So we know it couldn't be this one here, that's going to be a corner one.

It's not going to be this one down here and it's not going to be this one down here.

We've done that one, so now we've got seven left.

Now we can keep on going.

So pause the video and see if you can find the correct answer with these remaining four clues.

Okay great, so well done if you managed to work this out and let's just talk through the remaining clues.

So clue three, the Z-coordinate is less than the Y-coordinate.

Well, we can cross off this next one because the Z equals the Y-coordinate.

Same thing with this one, we can cross it off because the Z-coordinate equals the Y-coordinate again.

And are there any more that we can cross off? Well, we can't cross off the next one because the Z-coordinate is less than the Y-coordinate, and in fact, those are the only two that we can cross off.

The next one, the answer contains exactly two prime numbers.

So the prime numbers here that we could have are two or three.

So we need two of the coordinates to be two or three.

So we can cross of this one here, we can cross off this one here, and that is all.

So we're left with three candidates.

Clue number five, the X-coordinate is greater than the Z-coordinate.

Well, we can cross off this one here because the X-coordinate is less than the Z-coordinate there.

So we are left with two possibilities.

And we can use the final clue to help us find out which one it is.

So only one of the coordinates is a square number and the only square number that we can have is a one.

So we can cross off the two, two, zero, because there are no square numbers there, so we are left with the coordinates two, three, one.

And that is where our treasure is.

Really amazing job if you managed to do this yourself.

I hope you've enjoyed this lesson, I bet it was fun.

Next time, we're going to be looking at a similar problem involving a painted cube.

So really looking forward to doing that one.

Thanks so much for watching and see you next time.

Have a great day, bye.