Geometrical Concepts from Constructions, Models, and Investigations Summary

Tuesday, July 9, 2002

Jim passed out a triangle to each of us that had various points (A, B, or C) chosen as the origin. The other two points were (1,0) and (0,1). The task was to find the coordinates of D. We made a chart as follows.

            A coordinate (0,0)  B coordinate (1,0)  C coordinate (0,1)
A vertex                              1/4                  1/2
B vertex            1/4                                    1/2
C vertex            1/4               1/4

Looking at the chart, we realized that we could continue the pattern and fill in the values for the A coordinate of the A vertex, B coordinate of the B vertex and C coordinate of the C vetex. This did not affect any calculations, since anything times zero is still zero. We were then able to find the balance point on the triangles (point D) given imagined weights of A = 1/4 B = 1/4 and C = 1/2. In this way it was similar to Troy's vertex model of his triangle balance problem.

Troy showed us his Geometer's Sketchpad sketches to illustrate his project from last year (that he's continuing to refine) and explained about the Spieker center (center of mass for perimeter model). He also mentioned that any line through the Spieker point balances the perimeter and that any line through the midpoint of a side and the Spieker point also bisects the perimeter. Troy was able to demonstrate these concepts using his sketchpad dynamic sketch. We also talked some about the 9 Point Circle, Euler's line, and the line containing the Spieker point, Nagel point, centroid, and orthocenter.

We've also started to break into smaller groups to pursue problems to present as products. Steve planned to present some of his results during the Number Theory session tomorrow (Wednesday) morning. Troy will continue to polish and refine his project from last year. Jerry is pursuing the coffee cup problem and car guys concentric circle problem from Monday's work session. Joyce and Peg are contemplating 3-D models perhaps with applications to paper folding and origami. We'll try to pin all of this down more fully on Thursday.

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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.