While the meaning of ‘interior angles’ for a polygon seems intuitive, the meaning of a polygon’s ‘exterior angles’ can sometimes seem less intuitive to begin with. I tend to find that the go-to interpretation of exterior angles is often the reflex angle that covers the outside of the vertex. This seems like a more obvious way to interpret the term than it being an angle between the shape’s edge and an imaginary line.
A time that I remember the concept of exterior angles really clicking for me as a student was when trying to program a turtle robot to move in a way that traces an equilateral triangle. I can’t remember what robot it was (it was a long time ago), but it may have been this or something like it. After a period of bewilderment where I couldn’t understand why the thing kept nearly falling off the edge of the desk rather than coming back towards me, there was a lightbulb moment when I realised that it would have to turn 120 degrees in order to make a 60-degree angle inside the triangle.
These days, I find interior and exterior angles in regular polygons one of my favorite topics to teach. Firstly, there are great opportunities to make links between mathematics and computer programming, which seems particularly relevant today with the increased interest among children for coding. Secondly, there is a lot of fun to be had with problem-solving in this topic!
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Using Logo Programming to Introduce Exterior Angles
I’ve never worked in a school that owns any turtle robots to replicate this scene, but there are plenty of websites and downloadable software where pupils can do similar activities using coding. Two personal favourites for me are Python Sandbox and Logo Interpreter for their clean look. The images that follow in this post were made using Python Sandbox. In fact, some of my favourite maths lessons at school where when we visited the school’s computer room to use Acornsoft’s version of Logo.
If students are able to work at individual computers, then they could simply be shown the ropes and then tasked with exploring how to draw a square and then an equilateral triangle for themselves. However, when this isn’t possible, I’ve found that some great discussions can be generated in their normal classroom by posing questions for the class to discuss in pairs and answer on mini-whiteboards.
For example, the last few times I taught this to a Y8 class, I loaded Python Sandbox on the screen, briefly explained how it worked, and then displayed the code below on a PowerPoint slide.
Students wrote down the shape they thought it would draw (a square) and then discussed why they thought that. The key properties of a square were drawn out as they talked about how they could see that the arrow would move forward four times to draw four sides, each side would be the same length (100), and each angle would be 90 degrees. Following this discussion, we pressed ‘animate’ and watched as the square was drawn.
If students seem confident with Logo language, then coding loops could be introduced too, like in the image below. The number inside “range()” is how many times the instructions will be repeated. But for the remainder of this blog post, we’ll stick to writing everything out in full.
Admittedly, the 90-degree angle used to draw a square in this example above lures students towards a potential misconception. But it is this cognitive dissonance that leads to intrigue in the next task…
After showing students the animation of the arrow moving and turning to draw the square, I then displayed the slide below to the class. I told them I wanted to draw an equilateral triangle with side lengths 100 and this is what I have typed in so far. So, what number should I type in for the question marks?
Naturally, most of the class wrote 60 on their mini-whiteboards and then expressed surprise when they watched the arrow draw this…
The class was asked, “Why did the arrow do that?” Students discussed in pairs what they thought went wrong and why. It was helpful that the animation was set to its slowest speed so that they could see the arrow turning and notice that it didn’t turn far enough. Students seemed to grasp that the arrow’s turning circle was on the outside of the shape, so they should think about angles outside rather than inside.
They were then asked to answer the same question on mini whiteboards again, “What angle should go in the brackets?” There was roughly a 1:2 split in the class between those who held up 120 degrees and those who held up 300 degrees. So, we entered 300, pressed animate, and observed what the arrow did this time…
There were some chuckles and exclamations of “Huh? What’s it doing?” as the class watched the arrow slowly pivot almost in a full circle each time. The end result was that it drew a reflected version of the thing it previously drew. After giving students more time to discuss it in pairs, with many of them dragging their pens around the table to replicate what the arrow was doing, most got a good sense of what was going wrong. They were asked the same question on the mini-whiteboards one more time – the majority said 120 and were satisfied when the arrow drew the shape below.
They were then asked to consider why 120 degrees worked. What does that angle represent? Where did the number 120 come from? And how could we decide what angle to use if we drew a different shape? For example, what angles would we need to complete the code for this next shape?
After solving this problem, students usually seem to have grasped two points:
- They need to consider the angle that the arrow turns at each vertex. In other words, they need to imagine the line that the arrow would draw if it continued moving forward (which from now will be marked with a dotted line), look at the line they want the arrow to draw, and consider the angle between them. This is then referred to as the “exterior angle”.
- The exterior angle and interior angle at each vertex add up to 180 degrees because they make up half a turn (or a straight line).
Explaining these two points explicitly at this stage can also be a helpful moment of consolidation for those who have understood and a moment of clarification for those who are still unsure. These ideas can be reinforced further by posing questions, such as these next few.
The first allows students to directly apply what they have learned to a different shape…
I also like to pose questions in the opposing direction, whenever I can (see this previous post for more on this). So, this next question provides the exterior angles and asks students to work out what each interior angle must be.
Another key point for students to understand is that the exterior angles sum to 360 degrees for any polygon. I have found that setting the animations to ‘slow’ and allowing students to watch the arrow moving and turning gives them plenty of opportunities to see how it always starts and finishes pointing in the same direction – especially if using loops to code regular polygons (to make sure that final turn is included). Also, they can see that there is no other point in the journey where the arrow is pointing in the same direction as when it started. So, in total, it must make a single full turn while drawing the shape. I also like to illustrate this point by asking at least one student to stand up and re-enact the arrow’s movement, so they can see for themselves which direction they are facing at each stage of the journey. All this leads to the conclusion that all the exterior angles together make a full turn – in other words, the exterior angles always sum to 360 degrees.
This fact can make problems like the one below much quicker to solve…
Many students’ first instincts are often to calculate the two interior angles, calculate the missing interior angle, and then calculate its exterior angles. But they can then see how a quicker way would be to simply subtract the two exterior angles from 360 degrees instead.
So far, students have mostly played around with shapes that they are already familiar with. They have usually learned various facts about angles in triangles and quadrilaterals before this stage, so these new points about exterior angles just add to what they already know about these shapes. However, they can now use this new knowledge to deduce angle facts about shapes they are less familiar with…
The next post in this series will look at how students can make deductions about interior angles in regular polygons.
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If you would like to use or share any of the images from this blog post, feel free to use this Power Point to help you: Thinking About Interior and Exterior Angles (Part 1)
If you are new to turtle graphics and are interested in playing about it then you can use this guide to help you with Python Sandbox.
This post was the first of a three-part series of posts about interior and exterior angles. The other two parts can be found here:
- Thinking About Interior and Exterior Angles (Part 2): Deriving the Formula for the Sum of Interior Angles
- Thinking About Interior and Exterior Angles (Part 3): Different Ways to Create Problems with Angles in Regular Polygons
For my other blog posts, please visit my contents page.
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