The heart of this lesson is based on the act of connecting the simple visual process of lining up angles from a triangle and extended this to transversals, algebra and beyond. It is a topic that is accessible to everyone and challenging to everyone.
I like to start class with a list or table of common polygons and a question like, “what are the angle interiors of each polygon?” Then as we get those measurements up, I ask “why are the interior angles of each polygon that number?” Here I want my students to realize that the smallest number of triangles that “fit” or fill the area inside a polygon corresponds to the measure of the angles in the polygon. To help them reach this conclusion, I might drill a bit and ask questions like, “what polygon is missing on the board?” and “what is the smallest number of triangles that fit inside that polygon?”
As we discuss these observations, it is important to set these connections up in a table to help the class see that the interior angle sum = angle sum of smallest number of triangles that fill a polygon. I plan to use a Turn-and-Talk to present this concept to my students as a linear algebraic function. Usually, some of my students can infer that (n-2)180 = number of interior degrees. The fun part of this opening sequence is that this is all review. All we are doing is establishing a launching point for a better question, “why does a triangle have 180 degrees?”
My idea is that we can’t define other polygons in terms of a triangle unless we are absolutely sure that every triangle's angles sum to exactly 180 degrees. To run this experiment, I ask students to sketch any type of triangle on a piece of paper (and remind them of the three main types as defined by side length: equilateral, isosceles and scalene). They would then stack three sheets and cut out the three sheets to form three congruent triangles. It is important to demo this in front of the class. The idea is to arrange the three triangles into a straight line. It is critical that students record their work by sketching their work and results. They should also measure the angles in their triangles and label their diagrams accordingly. The instruction list should be simple, something like :
As students conduct their experiments, I circulate and push them to repeat the process with different types of triangles. I want them to think about how this could apply to any triangle, not just the three congruent triangles that they cut out. I ask questions like, “could you show this by sketching out a triangle with arbitrary angles, like angle a, b and c?” Suggestions like this spread the fuel of ideas for the summary, in which we discuss the meaning and extensions of the triangles and the exterior angle theorem.
Great Website: Triangles Have 180 Degrees
As we conclude this lesson, I want to show my students how the logic of transversals helps us to explain why the three angles must add to 180 degrees. I plan to walk the class through a proof by drawing any type of triangle and a line parallel to any of the three sides. When this construction is made, the side and parallel line are now the two parallel lines that will be cut by a transversal. The other legs correspond to a transversal cutting through parallel lines. The power of this activity is that students are acquiring the language needed to describe why a triangle has 180 degrees.
If you are unfamiliar with the proof, check this Khan Academy Video: http://youtu.be/OPG-9IFnJnI
I think it is important to use color in this part of the presentation. This helps students to visualize the argument as I lay out the proof step by step, pausing and asking students to rephrase and repeat the logic of the proof. As we discuss the proof, I make sure to record key student observations and to write explicit steps to the proof on the board.
When we complete the proof, I finish the conversation by discussing that true mathematicians always try to extend questions and make new discoveries.
I like to bring in a balloon or beach ball or inflatable globe and my bike pump to demo simple spherical geometry.
I use the pump to inflate the globe and show how a triangle on a sphere can have over 180 degrees.
Check this link for reference: "In Depth Analysis of Triangles on Sphere" and "Friendly intro to Triangles on Sphere."
I like this ending to the lesson since it points them towards the expanding nature of mathematics.