Bearings within regular polygons

Key Stage 3

Maths

Bearings

﻿Hello and welcome to today's lesson about bearings in regular polygons. It is the final lesson on bearings today so hopefully you've enjoyed this unit and hopefully, you'll enjoy today's lesson. For today's lesson, all you'll need is a pen and paper or something to write on and with. Please take a moment to clear away any distractions, including turning off any notifications. Finally, if you can, try and find a quiet space to work where you won't be disturbed, okay? When you're ready, let's begin. Okay so try this task. Now just before you start, I just want to tell you that this is an equilateral triangle, okay? That is really important for this question. All right. So now pause the video and have a go. Pause in 3, 2, 1. Okay, welcome back. Now, I'm not sure exactly what you've managed to work out, so let's go through it. So this is an equilateral triangle, so each of these angles is 60 degrees, okay? Now Carla has worked out B from C, she says that's 210 Where does she get that from? So that's B from C. Okay remember north, clockwise until we hit that line. So we want this angle there and then that angle there or a cool way that we could think about it is if I draw in my line and because this, that length there will be the same as that length there. Okay it splits it in half. So because it splits it in half, that angle will be 30 degrees here. Now that is a half turn, that is 180. So we have 180 plus 30, which gives us 210. Okay so that gets us to B, and then she's done A from B. So A, how to get to A from B. So that would be that angle there, which is just a right angle. So that's the 90 degrees and then how would do we do this one? Well, if that is 90 and that angle is 60, that angle there would just be 30, or you might have done it a different way. If you said that was 150 and they are co-interior angles, so they add up to 180. Okay that whole angle is 180 or just a part to that line there is 150, okay? So they are co-interior and they add up to 180. All right so if that's 30 and we want to go all the way around to there, well that's 360 subtract 30, which gives us 330. What's Zachy done? Well, he's gone the other way. He's gone from C to A, from A to B, from B to C. So he's used the fact that that is 150, okay? So remember 180 take away 30 is 150 to get to A. Then that is a 3/4 turn so that's 270. And then that angle there is 30 because that's a right angle, 90 take way 60 is 30. Okay, now if you add these up, you get 300 plus 330, 630. If you add these ones up, you get 300 plus 150, 450. That's strange. They're not the same. They're different. Why are they different? Do you notice anything about any of the pairs as well? So let's have a look here. We've got 210, that was Carla's first move. Zachy's first move, 150. What do they add up to? 90, 270. The second move, what do they add up to? What about the third move? Is there anything unusual going on here? And that's what I want you to have a little think about. All right? So you might want to pause and have a think about that. Find the three missing angles around B. So we're going to have to, in today's lesson, apply lots and lots of different angle facts. So we're just going to kind of build up our skills here and then we'll get chance to practise them and explore with these skills later on. What could I work out here? Well, I've got two parallel lines and this angle and this angle, they are co-interior. So if they're co-interior, what must they add up to? 180. So 180 subtract 105 gives me this angle here. Okay, can I work out that one? I'm not sure just yet, but I can work out this one. I can work out this one using angles in polygons, interior angles in polygons. What shape is this? How many sides are there? 1, 2, 3, 4, 5, 6, 7, 8, it is an eight-sided shape. So how many triangles would be in that shape. It would be six triangles, each with 180 degrees. So that would give me 6 times 180, which would give me a total of, okay? So that is the sum of all the interior angles, not just one. So to find what one angle is, I need to divide by eight because there's eight angles. So I get 135 degrees. Okay so now I've got that, that, and now I can use angles around a point, which add up to 360 to work out this missing angle. 360, I'll rewrite that zero, minus... Okay so 130 plus 70 is 200 plus 2 fives is 210, which is equal to 150. Okay. That was quite tricky, there was a lot going on there. So what I'd like you to do is to have your turn. So pause the video, have a go. See if you can work out the three missing angles around B. Pause in 3, 2, 1. Okay, welcome back. Let's see what you got. So this one should have been 100, they are co-interior, they add up to 180. Now this is a five-sided shape. So we need to work out the interior angles in a five-sided shape. Oops, sorry five minus two times 180. Okay 'cause there's always two less triangles than the number of sides. So that it's three times 180, which gives me 540. Now that is not the size of that angle, that is the sum of all these five angles. So I need to divide that number by five, equals 108. So that is 108. So then to find this angle, I need to do 360 subtract 100 plus 108, which gives me 152 degrees, okay? My pen is not having a good day today, I am very sorry. I'm having to concentrate very hard to try and write on this. Okay so hopefully, you got that, if not, hopefully you've found your mistake now. Okay, next one. What is the missing angle? Okay so this is like we did at the start. I'm just going to remind you then, because there was a lot of information then and you're going to use this skill for a similar question in a minute. So because it's regular, okay? And it's only because it's regular. Well, it would work for an isosceles as well, but it doesn't always work is the point that I'm trying to make. We can split this in half, which will mean that that is half of 60 degrees, which is 30. So this is 30 less than a half turn, which is 150 degrees. Okay so now I want you to use the principle of splitting an angle in half, bisecting the angle, to solve this one and you might need to know what the interior angle is of the shape. So pause the video and have a go. Pause in 3, 2, 1. Okay a bit tricky this one. So let's try and work out first what this interior angle is. Well, it's got eight sides. So how many triangles would be in the shape? 6, 6 times 180, which would give us this. Okay. That's the sum of the total interior angles, all 8 of them. So to find out one, because it's regular, I can just divide by 8, which gives me 135. So one of these angles is 135. What I'm going to do with this angle here is I'm going to split it down the middle. So that angle there would be 67.5, half of 135. Now this angle and this angle, they make a right angle, so they make 90 degrees. So to find the pink angle, I can do this. And that would give me 22.5, okay? 60 plus 20 is 80, 7.5 plus 2.5 is 10 so 80 plus 10, 90. Okay, next question. What is the missing angle? Well, first I want to try and work out what one of the interior angles is. Okay so let's try and work out what one of these angles is. Well it's a 1, 2, 3, 4, 5, 6, 7, 8, an eight-sided shape. So we can do the calculation that we did before. So we've got 8 subtract 2 to find the number of triangles, times 180. And from before, I remember that that divided by 8 gives me 135, okay? So that angle there is 135. But what is this angle? Can you see the right angle there? So that is 135 subtract 90. So 90 less than 135, which is 135 subtract 90, 45. Okay so that angle there is 45. I'm not asking for a bearing, I'm just asking for angle so that's why it doesn't need to be three figures here. Okay we'll use them in a bearings context shortly. Okay, your turn, all right? Pause the video and have a go. Pause in 3, 2, 1. Okay, welcome back. Hopefully you've worked out that one of the interior angles is 108. All of them add up to 540 so one of them is 108. So now we do 108 take away 90, which gives me 18 degrees. So that angle there is 18 degrees. Okay, we've got an independent task now and in this task, you are going to apply everything that we've discussed in today's lesson. So, pause the video to complete your task, resume once you're finished. So welcome back. Here are my answers. Pause the video if you need to mark your work. Pause in 3, 2, 1. Okay so just on two, on this one, the second one on number two, it was quite a long decimal and if you'd rounded it to a different number of decimal places, that's fine but to one decimal place, that was the answer and the same here, okay? So just mark your work there. Okay, well done. All right so now it's time for the explore task. Now is the time to apply everything we've learned over all of the bearings lessons and everything we've learned in today's lesson, okay? So I want you to explore any patterns that you noticed with bearings. You might want to think about clockwise and anti-clockwise, you might want to have a play around with that, looking at the sum of all the bearings, or you might want to think about what it would be for different shapes. So explore, play, try different things. So pause the video to complete your task, resume once you're finished. Okay and here is what I worked out. Now you might have worked out some different things, but I've got my bearings going clockwise 090, 180, 270, 000. My bearings going this way. And I also worked out the bearings going the other way. So they had gone that way around the square. And for the square, they were the same follow, that's . For the pentagon, they were different. That's strange I thought. They were different for the triangle as well. For the hexagon, they were the same. Pentagon has five sides, the triangle has three sides, odd numbers. The square has four sides, the hexagon has six sides, even numbers. What do you think for 7 or for 8 or for 9 or for 10? Do you think this pattern continues? This one, 540, 1080, that's double that. Is there anything going on there? Is there a kind of pattern there? Is there any connection between these numbers and the numbers with the triangles? Do we need to draw more shapes and work out those angles, those bearings to find that out? And I think we do. So that might be something you want to try now, okay? And I know I'm quite excited to try that. All right, so that is it for today's lesson and that is it for bearings. So really well done for all your hard work throughout this unit. So if you'd like to, please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak. Thank you very much for taking part and thank you very much for all your hard work, and I hope you enjoyed the lesson, thank you.

Bearings within regular polygons

Bearings within regular polygons

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UnitMaths / Bearings

Maths