1. A Historical Geometric Journey with GSP: Assessing Students’ Understanding Armando M. Martinez-Cruz, CSU Fullerton amartinez-cruz@fullerton.edu David Booze Troy High School dbooze1@earthlink.net Fernando Rodriguez Buena Park High School frodriguez@fjuhsd.k12.ca.us Presented at NCTM 2006 St. Louis, MO April 28, 2006

2. Overview of Presentation Welcome and Introduction, Class Project, GSP? Fernando: Pythagoras, Bhaskara, Garfield, Euclid and Similar Shapes. Armando: Some Applications of Pythagoras-- Quadrature of Rectangle, Golden Rectangle, Pentagon, Hexagon, Decagon David: An Extension: Pythagorean Triples Conclusions and Questions

3. Intro to GSP?

4. Pythagorean Theorem Bhaskara’s Proof Garfield’s Proof Euclid’s Proof Using Similar Shapes (instead of Squares) on the Sides of the Right Triangle

5. Constructions Using the Theorem Golden Rectangle A Square with the Same Area that a Given Rectangle, aka, Quadrature of Rectangle Pentagon, hexagon and decagon inscribed in the same circle. Actually, it is possible to construct a triangle with one side of the pentagon, one side of the hexagon, and one side of the decagon. And that triangle happens to be a right triangle.

6. Pythagorean Triples A Visual Demonstration of the Relationship Between Pythagorean Triples and Pythagorean Quadruples A Geometric Approach to Finding Pythagorean Triples An Algebraic Approach to Finding Pythagorean Triples and Beyond

7. Conclusions and Questions