Use these triangles to make regular polygons and deduce all of the angles!
Start with a puzzle and end with the proof for the interior angle of a polygon! This is a series of activities where you will solve puzzles and problems involving regular polygons. The purpose of each of these puzzles is to encourage you to use logic and reasoning to build on your existing knowledge of angles and shapes. At the end, the aim is that you 'prove' a formula for the interior angle of a regular polygon in different ways and, if you have time, create a wonderful expose of your work!
Start with this problem - can you make 8 regular polygons using all 42 of these triangles?
Here are the triangles for the Polygon puzzle pieces and the template you can use to help.
Then you will need the activities and information on this Polygon proofs activity sheet
The instructions and activity sheets above describe all the possible stages of this activity clearly. The following offers an overview and some thoughts to help prepare for the activity.
Part 1 - 42 triangles
Depending on your class, I recommend starting by just giving the students the sets of 42 triangles and seeing what they can do. This is a good puzzle for trying out different ways of thinking. Students tend to go for the congruent triangles that can used to make the hexagon and nonagon. After that, the level of thought required goes up considerably as students puzzle other ways of building regular polygons. (Note - some of the triangles need to be flipped)
But, then be aware that some of the students are likely to get a little stuck at which point giving the template will help things to move a long. Teachers need careful judgment here to differentiate accordingly.
The point of the exercise is that students are really thinking about the structure of regular polygons and the different ways they can be split in to triangles.
Part 2 - Particular problems
This section of the activity invites students to start looking at 'particular problems' that relate to the structure. There are diagrams and problems are broken down. Teachers might feel that some students could skip part 2 on move straight on to part 3 - you can always come back.
Part 3 - What are the angles
In this part, students should be putting together their experiences thus far to deduce some angles. They can be applying the approaches they have been using in part 2. Angle by angle, they can all be done. A key talking point though is the question below.
This is a terrific discussion and one worth having where you have time.
Part 4 - Proof
This is the part where we ask students to recognise the important difference between the general and the particular. Up to this point they have probably been jumping often between the two and so, in a sense, this is really about expressing the generalising they have probably been doing. The work done so far leads up to the idea that they can focus on three different ways of doing this. If students approach them in the order they are presented then there is a nice moment during the third proof where students often come up with a different result from the first two. This is a question of re-arranging the algebra to show they are equivalent. If we get to this stage then things have gone very well and there is a nice little 'wow' moment.
Part 5 - Presenting
I think this is a significant part of the exercise, but understand that it is time consuming. I am often envious of the time and effort that goes in to humanities projects (for example) and would like there to be more of that from my students. The trade off is that it can often seem that lots of time is used re-affirming mathematics that has already been done, BUT, it is about more than knowing how to calculate the interior angles of regular polygons. The exercise is about reasoning, logic, proof and communicating understanding. I want this to be a memorable experience so that that the thinking skills discovered or used can be applied in different situations. As such, there is no doubt in my mind that this is time well spent!
Here there will be some examples of student work that has been done on part 5 of this task where they are required to present some explanations and demonstrations of interior angle proofs. As shown in the task notes, students were given a free choice of medium for this task!
Here is a screencast of presentation that was done on this theme
This is a screencast of a project that was done on Scratch. There is an extra level of general thought required to program scratch to do this and I was really impressed with that!
Further submissions have included more posters, a comic strip and stop motion video......