Next year will be the 2nd year in a row I haven't taught geometry, but I had an inkling of an unformed idea for a possible exploratory activity. Whoa! Doesn't that sound like you just want to jump on it and use it the FIRST day of school (far far away from now).
I was in an Airport Shuttle and staring mindlessly out the window and saw a building wall made up of tessellated hexagons. Then I remembered how learned about angle sums of polygons in class by separating various polygons into triangles from one vertex and noticing a pattern [(#vertices - 2)*180].
Let's use the pentagon for an example of my future I-can't-use-it idea. Kids most likely don't know that there are 3*180 or 540 total degrees in all the interior angles of a pentagon. I was thinking of (for the pentagon group) having the kids use 10 pieces of spaghetti taped on 5 different scraps of paper to form 5 different angles. Then they would measure the angles and get a total sum of the 5 angles. Then they would try to manipulate the 5 taped down "2 sides and a vertex" scraps of paper to try to form a pentagon.
I'm thinking there would be 4 or so people in a group, and they would each have a different sum they would most likely come up with. Most likely a pentagon won't be able to be formed unless they happen upon a 540 degree sum.
Then maybe they could try again. Then maybe they could see what works and get that sum (having a human error discussion at some point for measuring mistakes). I'm also thinking there would be different groups with 4 sides and 6 sides (3, too??). In this way, they can see the tactical things that go wrong with angles that don't "fit". Then there would be a follow up summary and pattern.
Next post: things I could teach in 6th grade math ..... KIDDING