Students will use their knowledge that the angles of a triangle add to \(180^\circ\) to understand why the interior angles of a simple polygon add up to \((n - 2)180^\circ\). Students will learn both how to derive this formula and how to use it. Here's an excellent derivation of the formula. Students will also consider the special case of regular polygons, and solve problems involving them. One type of problem is to find the number of sides given one interior angle. Another is to find one interior angle given the number of sides. Here's another interesting type of problem:
Find the angle between each pair of regular polygons:
Here's the solution.
Next, students will learn what a regular tessellation is, and why there are exactly three regular tessellations. The proof involves nothing more than finding the interior angle of a regular polygon, and knowing that angles around a point add to \(360^\circ.\)
Next, give your students these challenges:
Transformations on a Pegboard by NRICH: For the first part of this challenge, we can move any of the pegs to make the right triangle.
Putting Two and Two Together by NRICH: Because NRICH states this problem is suitable for 7-11 years olds, but it references equilateral triangles and isosceles triangles, which are a grade 5 topic, these questions might be too easy for a 5th grader. What follows is how I would make the problem more difficult, for the students who aren't feeling challenged. There is only 1 free polyiamond for \(n = 2,\) but what about \(n = 3?\) 4? 5? 6? Drawings can be found here. The number of free polyabolos, for \(n \le 4,\) is also reasonable. I haven't found drawings for this, only the number of shapes for each \(n,\) which is here. For polydrafters, \(n \le 5\) is reasonable. Again, I couldn't find a drawing, but there's a listing of the number of shapes here.
TODO: Drawings for polyiamonds \(n \le 4\) and polydrafters \(n \le 5.\)
Conclude by leading this investigation: