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A Historical Geometric Journey with GSP:

Assessing Students’ UnderstandingArmando M. Martinez-Cruz,

CSU Fullertonamartinez-cruz@fullerton.edu

David BoozeTroy High School

dbooze1@earthlink.net

Fernando RodriguezBuena Park High School

frodriguez@fjuhsd.k12.ca.us

Presented at NCTM 2006St. Louis, MOApril 28, 2006

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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

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Intro to GSP?

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Pythagorean Theorem

• Bhaskara’s Proof

• Garfield’s Proof

• Euclid’s Proof

• Using Similar Shapes (instead of Squares) on the Sides of the Right Triangle

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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.

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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

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Conclusions and Questions