# Geometrical Concepts from Constructions, Models, and Investigations Summary

## Tuesday, July 2, 2002

We talked about how we enjoy using non-routine problems in our classrooms and mentioned some sources for these problems including Michael Serra's book, Discovering Geometry which has great problems at the end of each chapter. Also mentioned were UCLA Math Project, the University of Mississippi's Problem of the Week, the Math Forum, and Mu Alpha Theta's the Art of Problem Solving Volumes I and II.

We spent some time going through Philip Mallinson's activity Tessellating Around a Point with Regular Polygons and produced the chart of the 12 combinations of regular shapes using 3, 4, 5, 6, 8, 10, and 12 sides which will tessellate around a point. We spent a lot of time examining the relationship between the number of shapes that fit around a point and the sum of their reciprocals. For example, four squares fit around a point and 1/4 + 1/4 + 1/4 + 1/4 = 1. Also, 1 square, 1 hexagon, and 1 dodecagon fit around a point and 1/4 + 1/6 + 1/12 adds up to 1/2. It was amazing to note that if three shapes fit around a point, the sum was 1/2. If four shapes fit around a point the sum was 1. If 5 shapes fit around a point, the sum was 1 1/2 and 6 shapes around a point had a sum of 2.

We also spent some time working with Troy's project from last year on centers of gravity using a perimeter model and an area model. He explained Ceva's Theorem to us: For a segment that goes from vertex to any point on the opposite side of a triangle, three or more lines will be concurrent (intersect at a point) if and only if the product of the ratio of the pieces cut equals one.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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.