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MATH 5005 Geometry Summer 2006 Highlights Instructors: Bertha Orona, Don Gilmore, Bill Juraschek

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Sketching Geometric Solids Day 1 In the opening activity, participants made “geo- jackets” and nets for the candy boxes (Toblerone, Droste) they brought to class. Next, participants constructed rectangular prisms using snap cubes and drew sketches of their constructions. Bertha explains how to use color coding to make parts of a sketch connect to the real structure. Early student work with sketches.

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Cone Problem Day 2 Participants were given a paper cone and asked to determine its volume and surface area. Two participants present their formal solution to the cone problem. (The Toblerone container visible in the top picture was used in our opening activity.)

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Introduction to Geometer’s Sketchpad Day 3 Working with geometer’s Sketchpad would be a primary tool for exploring geometry content, and Bertha’s seventh graders use it extensively, so we arranged for them to introduce participants to its features.

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The Painted Cube Problem Day 4 Given a large NxNxN cube constructed of small cubes. If the outside faces of the large cube are painted, determine how many of the small cubes will have 0, 1, 2, or 3 faces painted. Participants struggled in their small groups. Eventually all groups arrived at the results shown. The formulas show the complexity of this problem, and the illuminating connections between geometry and algebra.

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Circuits and Instant Insanity Day 4 First we explored Euler and Hamiltonian Circuits, then we showed students how to model the Instant Insanity puzzle with circuit diagrams This was challenging for all, but worthwhile because of the extended interest is generated.

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Volume of Pyramid is One-third Area of Base Times Height Day 5 After calculating the volumes of several prisms, pyramids, cylinders and cones, many participants were still wondering about the use of one-third in the formula for the volume of pyramids and cones. We devised an activity in which participants constructed a special pyramid using straws and duct tape. Three of these identical pyramids can be placed together to form a cube. Using the straw pyramids as models, participants used Sketchpad to construct the net for the pyramid, printed three copies, taped the nets together to form pyramids, and then placed the three pyramids together to form a cube. The volume of the cube is (Area of Base)x(Height) where the base is the base of both the pyramid and cube, and height is the height of both the pyramid and cube.

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Sierpinski Pyramid Day 5 This culminating activity brought together many of the ideas we had been exploring on Days 1-4. Using marshmallows and toothpicks, participants first construct a regular tetrahedron and then use this unit to build larger pyramids. Lyndell presents her group’s findings about some of the relationships between the length of a side of the base of the pyramid and its surface area and volume.

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ELL Work with Frayer Model Day 5 Participants chose a geometric concept and then displayed its attributes, definition, examples and non-examples on posters. In the top picture, participants explain their work to the class. In the bottom picture, in response to instructor feedback, this group has added more illustrations of trapezoids in order to vary the irrelevant properties.

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Non-Euclidean Geometry Day 6 To do some work at the highest level of the van Hiele model we explored spherical and taxicab geometry. Bill used a large rubber ball and elastic bands to model a triangle in spherical geometry which has an angle sum greater than 180º. Participants worked through an activity exploring Taxicab Geometry that was taken from Teaching Mathematics in the Middle School. One discovery is that in Taxicab Geometry a “circle”-- the set of all points a given distance from a fixed point--is a Euclidean square.

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Using Similarity for Indirect Measurement Day 6 We spent a lot of time on mathematical similarity. It is an ubiquitous concept that involves proportional thinking, a kind of thinking that permeates the middle school curriculum. We have all seen the textbook problem where one calculates the height of a flagpole by using shadows and similar right triangles. We had the participants indirectly measure the heights of light poles around the school, set up sketches to show what they did, and use the proportions in similar triangles to calculate the unknown heights. We have found that actually doing the measurement project produces more lasting understanding of similarity and its role in indirect measurement.

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Transformational Geometry 1 Day 7 We began the exploration of transformational geometry by working with Miras. Participants had to explain how a Mira reflection/rotation works, mathematically.

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Transformational Geometry 2 Day 7 Don and Bill showed a giant snowflake created using rotations and reflections. A participant shows how she converted a construction done with a Mira into one on Sketchpad.

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The Earth View Problem Day 8 On the morning of Day 8, one of the instructors read a Wacky Question in the morning paper. We knew exploring the question and answer given could lead to some intriguing applications of geometry, so we planned some activities for the afternoon. Since we also wanted to do some work with proofs, we introduced some theorems about circles, and tangents to circles from an external point. Participants presented their proofs (which were critiqued by instructors). These theorems led to a natural model for exploring the question on Sketchpad.

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The Earth View Problem Day 8 To include some practical pedagogy, we had participants make booklets by cleverly folding three sheets of paper to yield a cover page and five pages on which to put their definitions and theorems related to circles. The finished booklets, were very popular. Finally, participants used Sketchpad to model the problem and explore the answers. In the words of one, “You can see virtually the entire hemisphere when you are one millimeter from infinity away.”

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Circle Folding Day 9 Start with a circle and fold it to form an equilateral triangle. Use theorems we have used to justify that this is an equilateral triangle. After folding the circle into a trapezoid, rhombus and regular tetrahedron, Bertha received “oohs and ahhs” for folding a truncated tetrahedron. She finished by unfolding the circle and asking for the area of one of the regions formed by the creases.

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Project Presentations Days 9-10 Participants chose a project from the booklet “101 Project Ideas for Sketchpad.” One of the most challenging was constructing tangram pieces that could be rotated and flipped. Julie and Katherine were proud of their work. Another project involved constructing a Sketchpad animation to show the path of the moon around the earth. The participant put in a lot of work on this challenging project, and learned a lot about geometry, but had not quite attained his goal at the end of the class.

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Tidbits Co-teaching was common. Don and Derrick needed an extension of the flip chart paper, and Bill provided it. Individual attention while working on Sketchpad was continual.

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More Tidbits These students were very pleased with the intricate snowflakes they constructed using paper folding. The pleasure of symmetry. Jesselyn had been totally immersed in a problem, struggling to understand her teammate’s explanation. Suddenly it made sense; she knew she had “got it.” Fortunately for us, her joyful expression lasted long enough to be captured by the camera, no posing necessary.

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