Simplifying Trig Expressions

This is a concept that I have always struggled to teach.   I believe in starting a concept by doing very informal math and slowly build up to the final product.   But with something like simplifying trig expressions I have never found a way to do that.   This year I tried something different and it turned out to be a lot better than anything I have tried in the past.

I started without giving any direction on what our final answer should be.   My initial goal was to get students comfortable with changing the expression using different identities and see what identities were on their reference sheet.     I gave each student a template with rows and 2 columns and a trig identity refence sheet that I numbered.

One column was for showing their work and the other column was for writing the number of the identity they used.  I also had students highlight the term they were changing.   I demonstrated the first problem.  They did the next example on their own.  I then had students do one step, then change papers with someone sitting around them, do one more step and so one.   For the first 30 minutes we just changed expressions using the identities.   The students had no idea what the final answer should be and thus, there was no way to get a wrong answer.  I felt like this helped with student engagement.  There was some moaning.  For example, “What is the point of this?”, “How is this going to be a test question?”, but nothing too drastic.  Once I felt like they had a good feeling for how to change things we switched to, The Trigonometric Identity Game.

The rules of the game were as follows:

I put students in teams of 4.  The first 2 minutes there was no talking with their team members.  After that, I allowed them to discuss with their team and come to consensus on their best answer.  I would walk around and check students’ answers and then award points.   Then I put a new problem on the board and repeated the process.

The minor switch of making simplifying a game where all answer are acceptable, just some are slightly better than others, made a huge difference.   Students didn’t get discouraged.  I never got the question, “How do I know I have the most simplified answer?”  Instead, students were actively trying to find a simpler expression.   Adding the -1 bonus point for solving the problem in the least number of steps had students reworking problems in different ways trying to find a more efficient method.  My job in between rounds became sharing the different methods the teams used,  as well as show other possible solutions.   This felt 100 times better than anything I did in the past which mostly was me doing a ton of example problems and kids doing a ton of practice problems.  I am curious to see if this can be used with other simplifying concepts.