In the morning precalculus sessions at #TMC13 we were asked to work with group members to develop a lesson, activity, assessment, etc to fit a topic that we found difficult to teach. My partner, @jlpaulen, and I chose to work on Trig Identities. We both felt that we find them a fun challenge and not too terribly difficult, but hard to teach.
What’s the Big Idea?
We determined that the struggle our students have with them is in the application of the algebra skills necessary to complete the proof. So for us, the big idea is simply algebraic manipulation using trigonometric functions.
What is so challenging for students?
The students have no idea where to begin or how steps flow from one to the next. There is evidence that students follow the mentality that they must know where to begin in order to finish the problem. Students are not receptive to the idea of try and possibly fail. While the students may have the ability to factor, distributive, substitute, etc in algebra, somehow applying these skills where a trig function is involved proves more difficult. I, personally, have had kids tell me that they tend to look at "sin" in a function and see 3 separate letters instead of one function. It makes them think they are dealing with a more complex problem.
Primary Identities to learn/memorize:
- Reciprocal Identities
- Quotient Identities
- Pythagorean Identities
- Sum and Difference Identities
Why were other identities not covered?
We settled on these 4 sets of identities because they seem to be the basis for many of the other groups. For example, odd/even identities can be taught using the sum/difference identities. (ie, sin(-u) = sin(u-2u).
- Make a trigonometric “card sort” where students are given pieces of paper with each step and reason on separate pieces of paper. Students are given a starting and stopping step. Their task is to put the "proof" in order. Students may eventually be weaned from being given reasons and eventually will be expected to write the steps and reasons entirely. Problems may also progress from easier to more difficult.
- Give the class a trig identity that can be verified in multiple ways. Divide the class into groups and give each group a large whiteboard on which to write their proof. Have each group present their proof to the class, then compare/contrast resulting methods.
- Another follow-up would be Kristen Fouss’ Trig Identity Matching Activity in Chapter 5 of her virtual file cabinet.
The card sort activity is focused on Pythagorean Identities (no sum/difference) in an effort to allow students the opportunity to practice these and develop habits before adding in sum and difference identities.
Note: The identities we used were from this website.