Involving Teachers in Mathematical Research

1
Involving Teachers in Mathematical Research

• David Booze
• dbooze1_at_earthlink.net
• Troy High School, Fullerton, CA
• Fernando Rodriguez
• frodriguez_at_fjuhsd.k12.ca.us
• Buena Park High School, Buena Park, CA
• Armando M. Martinez-Cruz
• Amartinez-cruz_at_fullerton.edu
• 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.