## Investigating Geometry Summary

### Monday - Friday, June 26 - June 30, 2006

On Monday, Art introduced the group to The Geometer's Sketchpad® (GSP), illustrating its dynamic nature. We did a number of exercises, covering the different methods of using GSP for conjecture and proof, noticing invariants (persistent relationships) when objects are moved, using measurement and calculation, tables, and so on. We discussed how GSP is "picky" (or if you prefer, mathematically correct). GSP has a lot of different kinds of functionality (for example, coordinate geometry), which often offers multiple methods to attack a problem; the functionality is always there, but is often hidden (and is often best kept hidden until you need it).

We practiced using GSP to prove by constructing auxiliary objects and discussed the general pedagogical value of the dynamic nature, since dynamic examples can show that relationships are not just true for a specific instance, but universally true for the situation. The ease of manipulating GSP constructions also means that one can go beyond paper-and-pencil in quantity and variety, and by allowing students to conjecture, makes the value of theorems clear through extensions. Different teachers start using GSP in different ways, some starting with it as a tool to make pictures or for classroom demos, but the tool achieves its full power when students explore. Joyce demonstrated a wide variety of functions using her "Ten Things I Love About Sketchpad" presentation and pointed us to a variety of nice resources on the Key Curriculum site.

On Tuesday, we reconstructed the virtual manipulative Aki showed us in Sketchpad, illustrating that constructed relationships persist even if the objects are hidden. We then constructed the four primary centers of a triangle (incenter, circumcenter, orthocenter, centroid) and made GSP tools to "record" the steps of the construction. We also examined scripts in GSP and thought about editing scripts. Joyce demonstrated the results of the nets in her presentations, a wide variety of beautiful polyhedra and space tessellations. We discussed published work from past Geometry Working Groups, the variety of places where such work started, and available resources to begin thinking about our own projects.

On Thursday, we went from paper-folding constructions to GSP constructions of a variety of conic sections. (We also discussed the equivalence of paper-folding and compass-and-straightedge constructions.) This process led us to some interesting GSP functions that connect to tangent lines, tracing, loci, and the connections between functions and constructions. This also brought up the dependency of constructions on the specific objects used (for example, lines vs. line segments). We discussed the use of this investigation with students of different levels, and the important insights for different students to take away. We concluded by discussing the specifics of selecting small-group project topics and the details of the presentation to the whole SSTP in the third week.

On Friday, we shared ideas for projects and began working in small groups on putting these together. These included a further exploration of midpoints of quadrilaterals into a lesson (working in the reverse of the "usual" direction to leave it more open for students), deriving the distance formula visually, an "electronic word wall" (perhaps using Sketchpad and Java Sketchpad so it could be online), using Zomes to investigate the Platonic solids (why only five? what is Euler's formula? what are duals? using soap bubbles to make minimal surfaces), investigating parameters in functions and transformations of the plane, and finding the surface areas and volumes of cylinders, spheres, and cones, and the relationships among them.

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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.