Lesson video

Hello and welcome today's lesson is about bearings on the Cartesian plan.

For today's lesson you'll need a pen and paper or something to write on with, and if possible, something that you can make a straight line with.

So it doesn't have to be a ruler but a ruler would be great.

Please now take a moment to clear well any distractions, including turning off any notifications.

If you can, please find a quiet space to work where you will not be disturbed.

Okay, when you're ready, let's begin today's lesson.

Okay, so I'd like you to have a go at this, try this task.

Using the diagram as a guide estimate the following bearings.

So I want you to pause the video and have a go.

Three, two, one.

Okay, welcome back.

So it's just an estimate we don't need to be super accurate.

And I'm going to kind of talk through my estimation process now.

So the first thing I'm going to do is plot this coordinate.

So we want to know roughly what this angle is here.

Now I'm going to draw in another line that is halfway between this access and this access.

So that's halfway.

So it forms an angle of 45 degrees.

That's kind of like our Northwest from last lesson.

So that is going to be less than 45.

I'm not exactly sure what it is, I think it's something around 35.

We can't work out the exact turn so we'd have to use something called trigonometry for that.

We're not getting into that just yet.

We're just kind of having a rough guess.

So something between 25 and 40, I would say is very sensible for an estimate.

Next one, five negative one.

Okay, let's draw a bearing in.

Oh, sorry, I didn't even put the extra zero in there, did I? I hope you didn't make that mistake.

Okay, so five negative one.

Now this is a much smaller angle than before.

So we should.

I don't know, I'd say that's maybe about something around 10 degrees.

Now that's 90, add another 10, which gives us an answer of 100 degrees.

And is that three figures? Yes it is.

Okay, next one, negative one, sorry, one negative five.

What is the same and what is different about this one and the last one? Well, that was five across one down, this time it's five down one across.

These two triangles, if you're imagining it like triangles, are the same triangles but they've been reflected.

So this angle here is going to be 10.

So if we know that that's 10, that's 180, 180 take away 10 which equals 170 degrees.

So we have a bearing of 170 degrees.

Negative two negative one.

Now this is less steep.

Sorry, it's steeper than that one.

So it's going to be a bigger angle.

And the other one kind of was like was over here.

I would probably say that this is at least double.

I probably said this is around 25 degrees.

Yeah, it's roughly about half.

If that's 45, that's roughly about halfway.

So I'd say something around 25 degrees is sensible.

So you don't have to be your exact answer, but something around that is sensible.

So going from North going clockwise to that access, it's 270.

But we are 25 less than that, so we have a bearing of 245 degrees.

Next one.

What's the same and what's different about that one and the last one? They're on the same line.

So they will have the same bearing.

Next one, negative two negative five.

Now the last one was four across two down.

So that roughly gives us the same angle as going two across and four up.

Okay, that would give us the same angle going that way.

See if you comparing this one with the last one, you can see that this angle there is going to be smaller.

So I'm going to say something roughly around 20 degrees.

So it's 20 degrees, less than a full turn.

So it turns 360 minus 20, which gives us a bearing of 240 degrees.

Okay.

Draw the North line in for points A, B, C.

The North line is always going to be parallel to each other.

So if that's going North, North will always go the same way for all the others.

And it doesn't matter if it's not a whole number coordinate, not an integer coordinate, the North line is still going to be going in the same direction.

The North line is always parallel.

And the parallel factor is really important.

So I want you to remember that, okay.

The North line is always parallel.

Okay.

What'd you think about this? Well, that's the coordinate.

From A it's going West, from B, it's going East.

They are not going to be the same bearing.

Bearing is all about relative position.

So if we work out this bearing, that's a three quarter turn.

So that is a bearing of 270.

Drawing our North line, work out that bearing.

That is a bearing of 90 degrees.

So a bearing of 090 degrees.

So these are different.

And they're different because with bearings, it's all about relative position.

So the coordinate three one is in the same place, but where it is with respect to you changes.

So you've got to consider with bearings where you are, where you are coming from.

Okay? This one was from A.

This one was from B.

And that's really important when we're doing bearings.

Where are you at? Where are you coming from? Okay.

Find two coordinates that are on a bearing of one, 90 degrees from the origin.

Find two coordinates that are on a bearing of 090 from the origin.

So from the origin, going North, going clockwise.

So that is that line here, okay.

We must always start North and go clockwise.

So that is there an angle of 90 degrees.

So we can have any two points here on this line.

And we can even have ones in between the whole number coordinates, there's loads of different ones we can choose.

So let's just choose this one, one zero.

I don't know.

Let's be funky and choose a three point five zero.

Okay.

Next one, 045 from the origin.

Well, that is that line here.

Okay.

Run back North, clockwise and it should be going through those corners that it's 45 degrees, that's halfway.

So we can choose any two points on this line.

I choose one one, and let's choose another funky one at two point five two point five.

You can just choose whole number ones if you want.

I just want to let you know by choosing funkier ones that we don't always have to choose integer coordinates.

Okay.

Three, 045 from A.

This time we're not at the origin.

This time we're at A.

So we've got this line here and we can choose any point on that line.

We can choose this one, two negative one.

And yeah, let's just choose that one, and three zero.

Okay.

135 from A.

So run back.

Around A, we start from North, we go around 135 degrees.

So it's any point on that line.

And that should be going through that little dot there.

So we could have two negative three.

And we could have three negative four.

And again, there's lots of points.

Points in between points, thousands, millions of points there in fact.

Infinitely many points there.

Okay.

315 from A.

So 315 from A, that'll be that line.

Trying to draw as straight as I can, obviously in a class we'd need a ruler.

So we could have this point and this point, this point and this point here.

Any two points on that line.

At negative two one.

And I don't know, negative three two.

Okay.

Now next one is a little different.

So we're at B, start North.

We go around 315 degrees.

So we've got this point here.

We've got the point at negative five two.

But what would come next? Well, it's off the grid we can't see it.

But this goes, one across one up.

So the next one that would go one across, negative six, one up, three.

So the next coordinate would be negative six three.

All right, that one was a little tricky, wasn't it? Okay.

So that was one across one up, one across one up.

So negative five to negative six, one up, two to three.

Find the coordinate on a bearing of 270 from A and 180 from B.

So 270 from A.

So we draw a North line in, three quarter turn.

So you've got this line here.

And from B we've got, start from North half turn.

Let me go there.

That is the coordinate that's on a bearing of 270 from A and 180 from B.

Okay, so we're going to do a bit of multiple choice now.

Which coordinate is not on a bearing of 315 from A? Pause the video if you need to.

Five, four, three, two, one.

Two four.

In case where we'd have, you should have worked out the bearing was this line here and you can see that the point two four is not on that bearing.

Okay, pause the video if you need to.

Five, four, three, two, one.

Okay, so it's going 45 degrees.

So it's going that way, which is one up one across, one up one across.

It has a gradient of one.

So it would be five plus one, six and five plus one, six and two plus one, three.

Okay, pause the video if you need to.

Five, four, three, two, one.

Okay, so we've got 225 from A, going that way.

And we've got 90 from the origin going that way.

So that will be that point there.

Okay, well don't know if you've got that correct.

Okay, so now I would like you to have a little bit of go at the independent task and come back here for answers.

So pause the video to complete your task, resume once you've finished.

Okay, these are not all the answers.

These are just possible answers.

If you manage to get points out also on this line or these lines, then your answers are probably correct.

And obviously I can't go through all the points 'cause there is so many that is actually infinitely many points if you're having decimals and everything like that.

But these ones, there should just be one unique answer.

So hopefully you got the same as me.

Okay, so the final thing for today is the explore task.

How many different coordinates can you find where two bearings pass? So if you were to, just for an example, take a bearing of 45 from the origin and zero zero zero from B, where would they pass? How many different coordinates can you find? Can you find any that are off the grid? Can you find any bearings where they have more than one coordinate in common? Okay, so I'd like you to pause the video and have a go.

Pause in three, two, one.

Okay, so hopefully you've paused it and had a go.

Now, I can't possibly go through them all.

So I'm just going to go for a few.

So if we had a bearing of 090 from the origin and zero zero zero from B, we'd have that point there.

If we had a bearing of 090 from the origin and 315 from B, we'd have that point there.

If we had a bearing of 090 from the origin and 045 from B, we'd have that point there.

Now, if you had the point of 045 from the origin, you have this line here, let's draw this in actually.

And then from B, if you had a bearing of zero zero zero you'd have this point here.

If you had a bearing of 315 from B, you'd have this point here.

And if you had a bearing of 45 from B, oh, you get this line and they are parallel.

They would actually never meet, which is interesting.

But there are ones that will meet off the origin, off the grid, sorry.

So if you took a bearing off 315 from C and 225 from A, you'd have a bearing meeting over here, and there'll be many others.

Now I kind of just want to talk about.

because there's so many for that first one, I'm not going to talk about them all.

Kind of mentioned how you could find one for the second one there.

But for the last one, that's really interesting.

And that's something I'm just going to have a little look at now.

A bearing of 180 from A.

And a bearing of zero zero zero from C.

They will have this line segment in common.

And now there's actually infinitely many points that in common on this line.

There's negative four zero.

There's negative four zero point one.

There's negative four zero point one three five.

There's infinitely many on this line.

There's also infinitely many that form on this line.

225 from the origin and 315 from C.

There are so many possibilities if you take into account decimals and even irrational numbers.

Okay.

So there's loads of points here and that's just interesting.

All right.

And it's something that I want you to think about and just remember that it doesn't always have to be one point but can sometimes be infinitely many points.

My little question for you then is, can there just be two points? I'll leave that one for you to think about.

Okay.

So thank you very much for all your hard work today.

I hope you enjoyed the lesson.

If you'd like to, please ask your parent or carer to share your work on Twitter tagging @OakNationalAcademy and #LearnwithOak.

Okay, so hope you enjoyed the lesson.

I look forward to seeing you next time.

Thank you very much.