Geometry Topics That Engage Students Friday, August 5, 2011 8:30 AM – 10:10 AM and 1 PM – 4:00 PM Sarah Mabrouk, Framingham State University, Organizer

There are a variety of geometry courses: some take an intuitive, coordinate, vector, and/or synthetic approach; others focus on Euclidean geometry and include metric and synthetic approaches as axiomatic systems; and still others include topics in Euclidean and non-Euclidean geometries and provide opportunities for comparisons and contrasts between the two.

§ What approaches are best?

§ What are particularly good topics with which to begin geometry courses?

§ What are some of the most enjoyable proofs to share with students?

§ What are the best ways in which to explore polyhedra, tessellations, symmetry groups and coordinate geometry?

§ How can we help students to develop the visualization skills for two and three dimensions as well as to help them to develop the mathematical reasoning skills that are important for studying/exploring/applying geometry at any level?

§ What are the best ways in which to compare and contrast Euclidean and non-Euclidean geometry?

§ How can we best convey the beauty of geometry to students?

This session invites presentations that address these questions as well as those that involve geometric topics from other courses including those for pre-service teachers. Presenters are welcome to share interesting applications, favorite proofs, activities, demonstrations, projects, and ways in which to guide students to explore and to learn geometry. Presentations providing resources and suggestions for those teaching geometry courses for the first time or for those wishing to improve/redesign their geometry courses are encouraged.

Session I – Friday, August 5, 2011, 8:30 AM – 10:10 AM

8:30 AM – 8:45 AM	Engaging Explorations in Geometry using Excel Deane E Arganbright, Divine Word University (Papua New Guinea) The spreadsheet is an excellent and engaging tool for facilitating student investigations of a wide range of concepts from geometry. This presentation presents illustrative examples of the analytic geometric use of Excel in algebra, geometry, calculus, linear algebra, and other classes. We present eye-catching animated models for the construction of pedal and inverse curves, evolutes, and similar topics, as well as the use of geometry in the classes listed above, and in the creation of attractive drawings for cultural designs and alphabet books. In the process of creating these models, students learn the underlying geometrical concepts through their implementations.
8:50 AM – 9:05 AM	Using GeoGebra to Improve Understanding of Proofs in Geometry William Schellhorn, Simpson College In this presentation, I will discuss how I use the free software GeoGebra to help students understand constructive proofs in geometry. My examples will include proofs in both Euclidean geometry and hyperbolic geometry. I will also discuss ways that GeoGebra can be used to help students form conjectures about properties of geometric objects.
9:10 AM – 9:25 AM	Voila! Proofs With Iteratively Inscribed Triangles Christopher Thron, Texas A & M University - Central Texas Designs with iteratively inscribed triangles can easily be created using dynamic software programs such as Geometer’s Sketchpad. Besides their visual appeal, these designs can be used in serious mathematical proofs that combine elements of classical geometry with the concept of limit. We demonstrate how to use iterated inscribed triangles to create a simple “Voila!” proof of the Euler line property. We will also show how iterated inscribed triangles can be used to characterize and locate any triangle’s Brocard points, and to characterize these points as points of concurrence of a special set of logarithmic spirals. These proofs are amenable to the discovery approach, and may be used in the classroom. All our demonstrations utilize “Compass and Ruler”, a freely-available Java applet that enables dynamic, iterative geometrical constructions.
9:30 AM – 9:45 AM	Proofs that Explain: An Example Margaret L Morrow, SUNY Plattsburgh Researchers in Mathematics Education often draw a distinction between a proof that convinces and a proof that explains. In this talk we will discuss a quintessential experience of this distinction for a result in the elementary geometry of the circle. Dissatisfaction with a proof that convinced, but failed to explain to the author’s satisfaction, led to an exploration on Geometer's sketchpad. This provided insight which enabled construction of a different proof which the author believes has more explanatory power. We will discuss the process and the proof.
9:50 AM – 10:05 AM	Visualizing Algebraic Surfaces Ivona Grzegorczyk, California State University Channel Islands We will show student projects involving classification of quadratic and cubic surfaces using singularities, symmetry groups, lines and curves on them as well as other geometric invariants. The visualization of these surfaces using modern software produce beautiful and unexpected images that serve as a motivation for students to work on semester long projects.

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Session II – Friday, August 5, 2011, 1 PM – 4:00 PM

1:00 PM – 1:15 PM	A Geometry Based Math/Art Course with a Studio Component Judith Silver, Marshall University Jonathan Cox, Marshall University Marshall University offers a four hour freshman honors seminar in mathematics and art, with emphasis on geometry. The beauty and usefulness of mathematics is enhanced by a studio component taught by an art professor. Topics covered include perspective, symmetry, mathematical themes in art, and studio skills.
1:20 PM – 1:35 PM	Geometry in an Historical Frame Ockle Johnson, Keene State College In this talk I will describe the course I developed that teaches geometry within an historical context. This approach has many positive features. Students read the first book of Euclid's Elements to review some basic Euclidean geometry. Students learn that important contributions were made by mathematicians from various civilizations. Students discover that geometry has developed over the years and that every development in mathematics, e.g., algebra, calculus, and abstract algebra, has provided new tools and raised new questions for geometry. Students realize that geometry is much broader than they thought and remains an area of active mathematical research and development.
1:40 PM – 1:55 PM	Analyzing Floor Plans: A Geometry Lab Emma Smith Zbarsky, Wentworth Institute of Technology I will describe a project that I developed for my geometry course. I collaborated with an architecture professor to select a number of floor plans designed by architecture students. Then I let the math students loose to prepare a lease space analysis applying basic concepts from two and three dimensional geometry.
2:00 PM – 2:15 PM	Symmetry and Shape: Geometry for Non-majors Penelope Dunham, Muhlenberg College What should a geometry course for non-majors look like? In particular, what topics will convey the beauty of geometry and, at the same time, attract students from the humanities who only want to satisfy their general reasoning requirement? My solution is “Symmetry and Shape,” a 100-level course that examines geometric concepts as it engages students with hands-on explorations and examples from art and nature. Although I originally designed the course to appeal to students from the arts, it has also been a popular choice for preservice elementary teachers and majors in biology, theatre, and history. This talk will address issues in designing the course, including topic selection and assessment options. I’ll list the major topics covered and give examples of innovative assignments, in-class explorations, technology-based labs, and available resources. I’ll also describe assessment components, including two portfolio projects: one focused on examples of symmetry from students’ environment, and another featuring original student art based on concepts studied in the course (culminating in a display for the campus Arts Week).
2:20 PM – 2:35 PM	Geometry Via Modeling Marian Anton, Centre College According to Euclid, if two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle. Starting from this example, we show how topological data analysis could reshape the teaching of geometry. In particular, we outline a course in elementary geometry aiming to engage students in learning via modeling.
2:40 PM – 2:55 PM	Using "Arts and Crafts" to Reinforce Geometric Concepts Kristen Sellke, Saint Mary's University of Minnesota My geometry course is a whirlwind tour through different geometries such as Euclidean, hyperbolic, analytic, finite, and transformational. Geometry is offered every two years so the students enter the course with a very diverse mathematical background. This presentation will examine a series of hands-on activities done throughout the course including constructions, paper folding and the creation of hyperbolic paper. We will discuss how my goals of the activities: to develop visualization skills for all students, to motivate proof-writing for second-year majors , and to give pre-service teachers examples of activities they can use in their future classes were met and look at student responses to the activities.
3:00 PM – 3:15 PM	Using Paper Folding to Explore Euclidean Geometry Carroll G. Wells, Lipscomb University Presentation Withdrawn (August 4, 2011)
3:20 PM – 3:35 PM	Kinesthetically Experiencing Geometry Todd D. Oberg, Illinois College With the increased emphasis on Transformational Geometry in the PreK-12 curriculum, and the continued need to study Euclidean Geometry, a rethinking of Geometry courses for preservice teachers may be necessary. One way of preparing future teachers, well as current teachers, is to combine Euclidean and Transformational Geometries into a single study rather than treating each as a separate topic. In this presentation, I will share some paper folding and Patty Paper activities that invite preservice students to more actively engage in the study of Geometry and also provide these students with opportunities to explore both Transformational and Euclidean techniques for creating proofs. In addition, some of these activities can be extended to explore ideas in Non-Euclidean Geometries.
3:40 PM – 3:55 PM	All Hands on Deck: In Praise of Toys Thomas Q Sibley, St. John's University Geometry students benefit from “playing” with geometrical objects. I have students use mirrors, basketballs, approximations of hyperbolic planes, both knitted and plastic, and other toys. These experiences help them develop valuable geometrical intuition and make conjectures. I will discuss how I have used hands on experiences to help students develop geometrical approaches to proofs and understand mathematical ideas.

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This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Thursday, August 4, 2011.