This blog post explores ways of teaching students how to find the sum of interior angles for any polygon, assuming they already know the sum of interior angles in triangles and quadrilaterals. For example, the sum of the interior angles in a pentagon:
Early in my career, I tended to take the most direct route possible to introduce students to the formula: sum of interior angles = 180(n – 2). This would usually be illustrated using the diagram below, which is probably the most efficient way to calculate the sum of interior angles.
There was nothing necessarily wrong with this method – after all, it is an efficient method, and the formula is usually the endpoint that I am aiming for. However, rushing to reach this endpoint as quickly as possible can sometimes bypass great opportunities for students to engage in some rich mathematical reasoning.
Inspired by the work of Akihiko Takahashi and my visit to Tokyo with the IMPULS Project in 2017, I’ve gained a greater appreciation for the benefits of allowing students to explore multiple methods for solving mathematical problems. It allows students to flex their mathematical creativity by applying prior knowledge to novel situations, which is a crucial ingredient of problem-solving. In my own experience, it also seems to reduce students’ perceptions of mathematics being a tightly closed discipline where every problem-type has a single solution strategy that must be learned and followed – which is certainly not the case. I have previously discussed this in early blog posts and would like to expand on it further in this post by considering different ways of calculating the sum of interior angles.
In recent years when teaching this, I’ve started by recapping angle facts about triangles and quadrilaterals (e.g., in a starter activity), before then displaying the problem at the top of this post on the board. Usually, setting it up by saying something like, “We don’t yet know what angles inside a pentagon sum to. But we do know what angles inside triangles and quadrilaterals sum to. I wonder if we can use those facts somehow to work out what the angles in this diagram add up to.”
The class is then tasked with splitting the pentagon up into shapes with fewer sides and then coming up with their own conclusions about what the angles in the pentagon might sum to. The last time I did this, students produced the following diagrams and calculations, which were displayed on the board.
It seemed that most students thought the interior angles summed to either 540° or 900°. They agreed that both can’t be true and were asked to consider which answer seemed more likely to be correct. They were reminded that they were trying to find out what the blue angles add up to and were asked to consider why the diagrams in the top row produced a bigger answer than the ones in the bottom row (which they discussed in pairs before reporting back). Some students suggested that it might be because some of the lines in the bottom row created new angles that were not part of the blue angles.
To pursue this lead, the class was then asked to consider which angles made up each 180° and 360° in each diagram. Doing so, they saw that some of the lines in the bottom row did create some additional angles, which wasn’t the case in the top row.
In particular, they deduced that new angles were created when lines intercepted at points away from the pentagon’s vertices. The main conclusion from this episode was that the better method for splitting up a polygon (in order to calculate the sum of its interior angles) was by cutting from vertex to vertex. That way, it avoids creating any new vertices.
However, just because this way is better, it doesn’t mean that the other methods are wrong. They are just incomplete! Each one could be finished off by subtracting the new angles that were made.
But even when drawing lines that start and end at the vertices, there is still a risk of new angles being created if the lines cross at any point inside the shape. I used to quote Ghostbusters whenever this happened and shout, “Whatever you do, do not cross the streams!” However, these problems can also be fixed too, by subtracting 360° for each new vertex created inside.
Learning this get-around doesn’t just help fix problems with crossed lines, it also opens up possibilities to explore other methods. For example, we could mark the centre of the shape and draw lines to each vertex to create five triangles. This inevitably creates another vertex at the centre, but we have a way to compensate for that.
If we assumed it was a regular pentagon, then all the triangles would be congruent isosceles triangles. This means we can calculate the sum of the two blue angles inside each isosceles triangle and then multiply it by five to get the sum of all the blue angles in the pentagon. This is one of the more conventional methods that I have seen in some math education books and websites – although usually for calculating a single angle in a regular polygon.
Another conventional method I have often seen is the one below. This depends on students having already learned that exterior angles sum to 360°.
One conclusion from this lesson could simply be that there are lots of different ways to calculate the sum of interior angles in a pentagon, which could also be applied to any polygon. Therefore, if they ever forget the formula (which we’ll get onto shortly), then they can always fall back on using their own mathematical reasoning to find their way again. But arguably the most important part of exploring multiple methods is what comes next: synthesising the methods to form generalisations.
If we look at the three more conventional methods (which don’t require us to subtract any additional angles), we can inspect their calculations and diagrams. One way could be to display all three on the board simultaneously, such as in the image below – which has combined each set of separate calculations into a single calculation. By doing this, students can see that even though the top two methods look very different in their images, their calculations are the same. We can also ask them to justify why these top two calculations are equivalent to the bottom one, which appears much simpler.
Alternatively, we could inspect each calculation line by line and determine what each number represents, before combining them into a single calculation. We could then ask, “What would happen if we changed the pentagon for another shape, such as a hexagon? Which of these numbers would remain constant? And which of these numbers would vary?” By then replacing the variable numbers with algebra, we can generalise each calculation to produce a formula. Crucially: the same formula each time.
At this point, students usually recognise that the last diagram is the most straight forward and explains the formula most clearly. Now, with the formula and pictorial methods established, students can apply what they have learned to find the sum of interior angles in other polygons. Furthermore, they consider the patterns that can be found when comparing polygons with consecutive numbers of sides.
…
This post was the second 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 1): Using Logo Programming to Explore Exterior Angles
- Thinking About Interior and Exterior Angles (Part 3): Different Ways to Create Problems with Angles in Regular Polygons
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 2)
…
For many more ways to find the sum of interior angles, I would recommend this post by @MrDraperMaths: Interior Angle Sum Derivations
For more about problem-solving in Japanese maths lessons, I would recommend these articles by Akihiko Takahashi:
- Characteristics of Japanese Mathematics Lessons
- Beyond Show and Tell: Neriage for Teaching Through Problem-Solving – Ideas from Japanese problem-Solving Approaches for Teaching Mathematics
For my other blog posts, please visit my contents page.
Pondering Planning in Mathematics 