Investigating Geometry Summary

Monday - Friday, June 29 - July 3, 2009

The first week of the Investigating Geometry working group began with introductions and then an interesting visual prompt. Jim discussed transformations and then showed us a point reflection using The Geometer's Sketchpad® but without telling us what it was. In small groups we brainstormed questions you could ask about this transformation.

We then had a rich discussion about the nature of this transformation, including whether it was best described as a rotation or as a reflection. This touched upon the value of generalizing the transformation to 3D. We next discussed the goals and constraints of the group projects and came to consensus that our investigations should focus on circles.

For the rest of the week we explored a number of rich circle problems that extended our understanding. We studied cyclic quadrilaterals and analyzed conditions for various types of quadrilaterals. We looked at traditional and more contemporary and inclusive taxonomies of quadrilaterals. We justified common constructions with compass and straightedge. We then looked at a variety of other figures that utilized circle theorems in novel ways. Most of us had our first exposure to orthogonal circles. We finished with a challenge regarding the Law of Sines in which we are to identify a segment associated with the length a/sin(A) for any given triangle.

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IAS/Park City Mathematics Institute is an outreach program of the School of Mathematics
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.