## Friday, December 30, 2011

### Two Final Problems

 Trig Problem 2
For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.

The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.

Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

 Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

 Have to love easy-to-draw memes.
Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.