1. Involving Teachers in Mathematical Research
David Booze
Troy High School, Fullerton, CA
Fernando Rodriguez
Buena Park High School, Buena Park, CA
Armando M. Martinez-Cruz
CSU Fullerton, CA
Presented at
Key Curriculum Press
Sketchpad User Group
St. Louis, MO
April 27, 2006

2. Context
Typically geometry courses for mathematics teachers use a lecture approach with little time for independent study. We report on our approach to engage mathematics teachers in research. This approach consists of the use of dynamic software to investigate a problem, and some derived problems, which are obtained by changing the condition(s) of the initial problem. Examples of teachers’ work will be presented.

3. Let ABCD be square. Let H, I, J, and K be the centroid of triangles ABC, BCD, CDA, and DAB respectively. Prove that HIJK is a square.

4. Same Idea with a Parallelogram

5. Same Idea with Any Quadrilateral
Aha… they look similar… but now we have a better idea of the way this works… a rotation!

6. One More…
Consider P an interior point of a quadrilateral. Use two consecutive vertices and P to determine a triangle and find the centroid.

7. Another Variation
ABCD a rectangle, and diagonal AC. Construct the perpendiculars to AC from B and D. Let I, and J be the intersections with the AC. Construct the centroids of triangles ABI, BIC, CJD, and JAD.

8. What to investigate…
The original problem suggests to look at the “inside square”. However, here are some options:
Is there a relationship between the areas of the figures?
Is there a relationship between the perimeters of the figures?
What about other points (as incenter, circumcenter, orthocenter or a combination of these)?

9. Reflections and Conclusions
Students feel the final project is “impossible” at the beginning of the course.
The final project gives the students a sense of ownership of the mathematics.
The project promotes an inquisitive mind.
Several students have co-presented at national conferences their result.
This project gets at the core of the teacher-researcher issue.