PreCalculus Session - Trig Identities Card Sort etc

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).

Activity ideas

- 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.

TrigIdentityCardSortPythagorean (1)

Parent Function Card Sort

This week I spent in an inordinate amount of time working on materials for Geometry, but managed to work in some time for PreCalc as well.  Early in the year we spend time going over the 12 Parent Functions.  Students are then taught how to create new functions using the four basic operations, compositions, and transformations, as well as how to analyze these functions.  One of the problems my students have previously had with these was remembering how the parent function behaved in order to determine how a transformation of that function would affect the domain, range, etc.  In an effort to curb this problem, I am creating a card sort using the pages below.

My goal is to print the graphs, domain and range, etc on separate colors of cardstock.  Students will not be given the entire set of cards at one time.  Rather, they will be given the pieces as they are learned in class. I am scheduling time in class as these are covered to review the previous day's information, before adding new material.  By the time the entire analysis is taught, the students will have practiced the sets enough times that they've memorized, or can quickly determine by sight, the analysis of each parent function. Hopefully, this will also improve their speed in analyzing transformed functions throughout the remainder of the year.

Parent Function Card Sort

Triangle Centers

I am in the process of working through a series of lessons on triangle centers.  So far, I've made it through the perpendicular and angle bisector theorems, as well as perpendicular and angle bisectors of triangles.  I am posting what I have accomplished here in an effort to get input on whether these seem to be too difficult.  Are my expectations too high?  My students are average level sophomores.   I am wavering on including more basic skill level types of questions along with the problems currently on the worksheet.

I have considered updating parts of it by incorporating some sort of technology investigation using either TI-84s or 92s, but would have to incorporate some sort of training on how to use the Cabri environment before looking at triangle centers.  I think this would be especially useful in portions of the investigation into angle bisectors of a triangle.

I thank you in advance for any help or advice.

Perpendicular Bisector Theorem Investigation

Angle Bisector Theorem Investigation

Perpendicular Bisectors of a Triangle Investigation

Angle Bisectors of a Triangle Investigation