Day 88: Looking for patterns

I began class with a derivation of the angle relationships that appear when a transversal intersects a pair of parallel lines. We started with a new, horizontally drawn rectangle on a graph so that the distances would be easier to eyeball. Just as we did when initally making conclusions about rectangles, we solved for the measure of an acute angle formed by the diagonal using a tangent ratio. From there, we can complete the right triangle to solve for a third angle. Then use the angle addition postulate (which still provides a perplexing amount of trouble for students) to solve another angle, and continue to solve for angles using linear pairs and vertical angles where needed.

When it's all done, the picture is a mess and students are stressing that they'll have to replicate all 15 (or however many) steps there were for every problem. "Wouldn't it be great if we had some shortcuts to follow?" I asked. "Do you notice any commonalities that we could look for again in the future?" Some angle pairs jump right out, but even the others aren't hard to see with a little effort. 

So I created 4 copies of the framework of the picture (just the parallel lines and the transversal) and identify one pair at a time, naming them as I go along. 

Students then had ~20 minutes to work on U6 WS1 and were told that it would be checked (for a grade now) on Monday.