Working with triangles

The next unit in geometry covers triangles. I started by using a graphic organizer to see what students already know and correct any misconceptions. The organizer lists names of triangles (scalene, isosceles, equilateral, obtuse, etc.) down one side and properties (no congruent angles, 2 congruent angles, 3 congruent sides, 1 right angle, etc.) across the top. I had students fill out what they could, discuss in groups and then discuss as a class. I asked the class to complete this in pencil as I knew there would be corrections.

This led to some good questions and discussions about whether certain properties could exist. We worked through these questions and misconceptions and then used the information to either name or create described triangles.

The next class started by working through a couple more problems using naming to identify triangles. This class turned into a long, drawn-out affair because students couldn't find the side lengths of a triangle given its coordinates. I was baffled by this. I finally drew a single line segment on the graph and asked how they would determine the length of the segment. Students readily said that you would create a right triangle and use the Pythagorean theorem. It became apparent that students did not recognize triangle sides as line segments. Once we got over this hurdle we were able to complete the remainder of the practice problems (with some struggles on computation). Unfortunately, this took the entire class.

Today we worked through proving the interior angle sum theorem for triangles. I started with three progressive problems: the first gave to values and then asked the measure of the third angle, the second gave one value and two expressions for angle measure and asked for the value of x, the third gave three expressions for angle measurement and asked for the value of x. I anticipated that moving from the first to the second would give some students pause, which it did. Once they realized that the same interior sum property held, they were able to move on.

I then asked students what properties or theorems could be used to help prove that the interior angle sum theorem was true. After briefly discussing in groups we did a share out. Many of the groups thought if we could somehow relate the interior angles to a straight line it would help. (Many students were thinking along the line of unfolding the triangle.) One group suggested using a protractor and compass. A couple of groups thought the triangle angles all had to be less than 180^o and that perhaps that could be made use of.

I drew a line parallel to one side. I could relate this to their thinking about somehow relating the angles to a straight line. I instructed them to look at the angle relationships when a transversal crosses a pair of parallel lines. How are the angle relationships connected.

This again was a struggle for the class. I kept reminding students to draw things out and make connections. Finally, a couple of students drew out the triangle, extended the sides to lines and then drew a parallel line. They started to mark congruent angles and saw the alternate interior angles connected to angles in the triangle.

I wrote out the first few steps in the proof and asked students to try and finish a two-column proof based on this start. It will be interesting to see if any are able to complete the thinking.