I did this kinda fun hands on intro to the angles in a regular polygons earlier this year and I wanted to share. It was inspired/thieveried by an Illustrative Mathematics lesson that I can’t find on the internet now (I think their curriculum is about to come out, which I’m excited about, but maybe some of their stuff online going away) and this blog post from Jennifer Wilson, so nothing new, but I figured I’d amplify and give my own thoughts.

1. Fold the Octagon:
   • Take a square piece of paper, and halve it by folding one edge to the other edge across the way, unfold, and do it the other way too. Then halve it along the two diagonals too.
     
   • Then fold all the halfway lines between those lines. Easiest way to do this is fold along the current lines until you get one of those eighth triangles, and then fold the folded edges to each other (see picture).
     
   • Unfold. Then find the points indicated in the diagram and make a fold on the line between those two points (my diagram is a bad construction of this, they shouldn’t be exactly quartering the top).
     
2. Any conjectures about the shape? We talked through some of these and talked through why they might be true. Good reasoning involved the fact that when you fold something onto something else, it makes it congruent.
   • It’s an octagon.
   • It’s a REGULAR octagon.
   • There are four kites in the figure (can you see them?)
   • There are 8 isosceles triangles.
   • The last fold we did made a 90 degree angle with the other fold.
3. Now physically label the measure of all of the angles. Like, with a pencil and not your brain. Justify your thoughts.
   • This was interesting because there seemed to be two ways to go:
     • Some started with the corners. Since that angle is folded in half, those are two 45 degree angles. And the last fold makes two 90 degree angles because the angles are folded on top of each other (congruent) but also along a line (supplementary) so have to be 90. Then the other angle in those corner triangles is 45 degrees…. and go from there.
     • Some started in the middle. Since we folded all 16 of those central angles on top of each other, they have to be congruent, and since they add up to 360, they need to be 22.5 degrees. Then lots of places to go from there either with isosceles triangles or right triangles.
   • After kids spent 10 minutes or so labeling on their own or with a partner, we took turns sharing some ideas. Some kids talked through their thinking and we all gave critiques. It was a good switch between individual-full class modality too.
4. Now, draw a STAR where there is an exterior angle and a HEART where there is an interior angle.
   • We had just learned what these and had learned how to calculate them so I wanted to see if they could find one on this complicated diagram. This was harder than I thought it would be, which meant it was a good use of time!
   • Do the measures of these angles match with the equations we figured out to calculate them? (Yes! 45=360/8 and 135=180-45=180(8-2)/8).

This took maybe 40 minutes (I can’t tell from my lesson plan if that’s right), but was great! It was physical and exploratory and was full of fun geometrical arguments that are based in transformations, but also had a nice, concrete, angle-labeling component for kids who prefer numerical lessons. This gets my 😀 of approval!!!