Article Summary

SUsing Technology to Unify Geometric Theorems About the Power of a Point    PDF

2011, Vol. 21, No. 1, 11-21

In this article, I describe a classroom investigation in which a group of prospective secondary mathematics teachers discovered theorems related to the power of a point using The Geometerís Sketchpad (GSP). The power of a point is defines as follows: Let P be a fixed point coplanar with a circle. If line PA is a secant line that intersects the circle at points A and B, then PA*PB is a constant called the power of P with respect to the circle. In the investigation, the students discovered and unified the four theorems associated with the power of a point: the secant-secant theorem, the secant-tangent theorem, the tangent-tangent theorem, and the chord-chord theorem. In our journey the students and I also discovered two kinds of proofs that can be adapted to prove each of the four theorems. As teacher educators, we need to design learning tasks for future teachers that deepen their understanding of the content they are likely to teach. Having a profound understanding of a mathematical idea involves seeing the connectedness of mathematical ideas. By discovering and unifying the power-of-a-point theorems and proofs, these future teachers experienced what it means to understand a mathematical theorem deeply. GSP was an instrumental pedagogical tool that facilitated and supported the investigation in three main ways: as a management tool, motivational tool, and cognitive tool.

About the Author:
Dr. José N. Contreras,, teaches mathematics and mathematics education courses at Ball State University. He is particularly interested in integrating problem posing, problem
solving, technology, history, and realistic mathematics education in teaching and teacher education.

Last modified: 30 July 2012.
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