## Geometry Summary

### Monday - Friday, June 27 - July 1, 2005

Monday, June 27, 2005
For the first half of the class, we discussed some of the possible "tools" we could use do our geometry work over the next three weeks. These tools are: Some of the proposed content or topics were:
• 3-Dimensional Geometry
• Spherical Geometry
• Transformations and Symmetry

We spent the better part of an hour working in the computer lab with Geometer's Sketchpad, helping the participants work with the software, and trying to assess who knows what about the software.

Our general feelings after today are to give the participants plenty of opportunities to work with each of the tools before deciding on which topic the participants would like to approach.

Tuesday, June 28, 2005

Our Geometry Working Group Representative to Carol's Wednesday Morning Breakfast Club is Melissa Garza. Kudos to Melissa for agreeing to take on this arduous task. For her efforts, Melissa will get a free breakfast!

We began today by playing around with Zome Tools. After we became familiar with them, we extended a problem from the 11:00 hour by building as many isosceles triangles as we could using the blue struts. This prompted a discussion of the angles in certain isosceles triangles. In particular, we explored this triangle:

From this triangle, we concluded that the following ratio held:

This is also a connection to the morning sessions, for this is the golden ratio.

We next spent some time talking about constructing these things in Sketchpad. One of the things we did was to create a Hide/Show button.

To CREATE A HIDE/SHOW BUTTON.
Select the things you want to Hide.
Edit Menu - Action Buttons - Hide/Show

We also talked about creating tools.

Thursday, June 30, 2005

We worked on building straw models of different solids and writing questions that went along with the models we built.

Theresa built a hexahedron using two tetrahedrons.

1. Where is the center of gravity and does it have anything to do with the intersections of the centroids?
2. Do lines connecting the centroids meet? What about the other points of concurrency?
3. What happens to the height of the solid as the size of the triangular faces increase?
4. How would the center of gravity change? If the solid was not made of equilateral triangles?
5. What are some types of cross section areas?

Mike built an octahedron in a tetrahedron.

1. How do surface area and volume scale with uniform linear growth?
2. What is the surface area of the different shapes?

Ginny built a tetrahedron.

1. Can this model be used to show how both surface area and volume change as the shape is scaled?

David built a five tetrahedron flying disk.

1. What is the surface area?
2. What is the volume?
3. Do the tetrahedrons meet at the center of a circle?
4. What is the height of the pentagonal pyramid?

Judy built three tetrahedrons of different sizes.

1. What are the relationships of the lengths, surface area, and volume?

Josue made a cube illustrating a face diagonal and a space diagonal .

1. How does this demonstrate a 45-45-90 triangle?

Kelly built a square pyramid.

1. Using the altitude, find the area of each face.
2. How does the volume change with changes of scale?

Jim built an octahedron.

1. What is the height of one of the square pyramids?
2. What is the volume of the 2 square pyramids?
3. What is the measure of each dihedral angle?
4. What is the volume of each triangular pyramid?
5. Generalize your results for different size straws.

Friday, July 01, 2005.
Steve gave a brief demonstration of Cabri 3D using some of the figures constructed the previous day.

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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.