Geometry Topics that Engage Students Friday, August 6, 2010, 1:00 PM - 4:55 PM and Saturday, August 7, 2010, 1:00 PM - 4:15 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 effective?

• What are successful topics with which to begin geometry courses?

• What are some enjoyable proofs to share with students?

• What are effective ways in which to explore polyhedra, tessellations, symmetry groups and coordinate geometry?

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

• What are effectual ways in which to compare and contrast Euclidean and non-Euclidean geometry?

• How can we convey the beauty of geometry to students?

This session invites presentations addressing these questions as well as those involving geometric topics from other courses including those for pre-service teachers. Presenters are encouraged 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/suggestions for those teaching geometry courses for the first time or for those wishing to improve/redesign their geometry courses are encouraged.

• Session 1, Friday, August 6, 2010, 1:00 - 4:55 PM

1:00 - 1:15 PM	Activities to Enliven a Course on Euclidean and non-Euclidean Geometries Sarah J Greenwald, Appalachian State University Creative activities and projects can enliven a classroom and help students develop an ownership of the course material. In this talk we will discuss the creation and implementation of the following activities in a course on Euclidean and non-Euclidean geometries. Axiom Systems and Wile E. Coyote http://mathsci.appstate.edu/~sjg/class/3610/wile.html Historical Timeline Project http://mathsci.appstate.edu/~sjg/class/3610/timeline.html Euclidean and Spherical Polyhedra http://mathsci.appstate.edu/~sjg/class/3610/platonicsolids.html Parallels and Connections Among Geometries http://mathsci.appstate.edu/~sjg/class/3610/parallels.html We will discuss the student reaction to the course before and after the implementation of these projects and classroom activities, and the benefits and challenges of including them in the course.
1:20 - 1:35 PM	Finite Geometries and Games Kay Ellen Smith, Saint Olaf College Finite geometries provide a good setting in which to introduce axiomatic systems and illustrate the meaning of independence of axioms and model of a system. In this talk we describe two activities in which games are used to explore models of finite geometries. In the first activity students find models of a plane configuration (a type of finite geometry) that contain nine points with three points per line. After discovering three non-isomorphic models, students play tic-tac-toe on the models to help them better understand the differences in structure of the three models. In the second activity students first play the game of SET ( http://www.setgame.com/) and then investigate questions about the finite geometry underlying the game such as "How many points are there?" and "How many lines pass through a given pair of points?"
1:40 - 1:55 PM	Excursions on the Sphere Kristen Schemmerhorn, Dominican University Exploring spherical geometry can help students get a better understanding of the properties of Euclidean geometry and hyperbolic geometry by comparing the different geometries. Using the Lenart sphere, students have an opportunity to discover properties of spherical geometry. Along with discovering properties of lines and triangles and comparing them to Euclidean and hyperbolic geometry, spherical geometry can also be used to see why the betweeness axioms are important for the exterior angle theorem. Some guided discovery activities on the Lenart sphere will be presented along with a discussion of what students should learn as a result of the activities.
2:00 - 2:15 PM	Engaging Students in Learning about Scaling Davida Fischman, CSU San Bernardino Scaling involves fundamental geometry concepts, and is one of the most difficult topics for students to appreciate and understand. As a result, they often "check out" and passively await the next topic. We will describe a progression of activities that kept one class of liberal studies students engaged and actively involved in developing an understanding of scaling and its applications.
2:20 - 2:35 PM	Geometric Art and Algebraic Surfaces Ivona Grzegorczyk, California State University Channel Islands We will show geometric activities based on the use of software for stimulating interest in 2D and 3D geometry. We include interesting mathematical images and surfaces from arts, science and architecture and meaningful artistic creations designed by students.
2:40 - 2:55 PM	Projective Geometry -Visualizing proofs and interpretations in Euclidean Space Xiaoxue Hattie Li, Emory & Henry College The study of projective geometry, besides its own beauty and elegance, provides a way of gaining perspectives on other modern geometries. Euclidean geometry is a special case of projective geometry in the sense that the group of Euclidean motions is a subgroup of the group of projective transformations. This presentation will introduce two undergraduate research projects conducted in and after a modern geometry course. The first project explored the proof of Desargues' theorem in both two and three dimensional projective spaces through making 3D models of different constructs of the theorem. This has greatly engaged students in learning geometry since they could visualize the proofs and it stimulates their creativity. Our second project discussed the relationship between projective geometry and Euclidean geometry by interpreting this theorem, which takes place in projective space, to Euclidean setting. We provided two corollaries for Desargues' theorem when it is valid in Euclidean space.
3:00 - 3:15 PM	Non-euclidean geometry across the '7 grade / major' spectrum Jack Mealy, Austin College In this talk we first outline various subcategories of Geometry, in particular emphasizing the broader (and perhaps less standard) definition and scope of the subject. Then attention is given to the levels of difficulty in the study of these various subcategories. Finally we discuss efforts to bring specific proper subsets of these subcategories of Geometry into the curriculum of three phases of education: for Math majors, for the general education college student, and for students in High School / Middle School. Specifically: Student projects from my upper level Geometry course will be discussed. Specifics from my January Term course for general education students will be discussed. Finally, we discuss the beginning of a new effort, the OTP (Outside The Plane) project, to introduce some of these ideas even more widely, i.e, to a segment of students at grade levels 7 through 12. Some specific examples will be included.
3:20 - 3:35 PM	A Feuerbach Refresher Len Smiley, University of Alaska Anchorage A digest of classical approaches to Feuerbach's Theorem on tangencies to a 9-point circle provides perspectives and toolkits often stimulating to prospective secondary educators.
3:40 - 3:55 PM	An Inquiry-Based Approach To Middle-Level Geometry For Preservice Secondary Teachers Diana White, University of Colorado Denver We discuss course methodology and outcomes of a course for pre-service secondary teachers that involves an inquiry-based approach to middle-level (grades 4-9) geometry. This course was piloted in Summer 2009 and taught a second time in Summer 2010. Secondary teachers in some states are certified to teach grades 6-12, yet often we attend only minimally to the mathematics in grades 6-8. However, the mathematical knowledge needed to teach grades 6-8 is substantially different from that of the upper grades, and not traditionally taught in a mathematics major. This course arose out of a revision of our pre-service secondary math teacher preparation program to better align with the Conference Board of the Mathematical Sciences recommendations on the Mathematical Education of Teachers.
4:00 - 4:15 PM	Developing Visualization Skills through an Exploration of Platonic Solids Using Technology and Traditional Methods Cheryll Elizabeth Crowe, Eastern Kentucky University This paper describes a project implemented in a geometry class for elementary and middle school pre-service teachers. The purpose of the project was to develop visualization skills from 2D to 3D and make connections to surface area and volume through an exploration of Platonic Solids. Through the project, students also expanded their knowledge of Euler's formula and ascertained the history of Platonic Solids. The project commenced with an overview of constructing nets for Platonic Solids using traditional methods (compass and straight edge) and technology (Geometer's Sketchpad and GeoGebra). Pre-service teachers constructed nets for each Platonic Solid by hand and with technology; then they calculated the surface area of each figure. Following this task, nets were folded to create three-dimensional renderings of the Platonic Solids, and students determined the volume of each figure. At the conclusion, pre-service teachers used their 3D Platonic Solids and an activity from the National Library of Virtual Manipulatives (NLVM) to explore Euler's formula and examine the history of Platonic Solids. In this talk, the presenter will discuss the project, show samples of constructed Platonic Solids using traditional methods and technology, and demonstrate the NVLM activity exploring Euler's formula.
4:20 - 4:35 PM	Informal Geometry for Aspiring TV/Film Directors and K-8 Educators Lucy Dechene, Fitchburg State University The Informal Geometry class we developed for K-8 pre-service teachers became extremely popular with Communications Media majors long before it became a requirement for education majors. The course is organized around 19 discovery labs and topics from Geometry: An Investigational Approach (O'Daffer and Clemens). The underlying theme of the course is symmetry, although topics from Fractals to Topology are investigated. A fair amount of time is spent on polyhedra. Topics and labs will be discussed in this presentation as well as the varied topics students have used for the 15- page research paper/visual semester project. We encourage as projects for TV/film majors the analysis of film scenes from favorite movies for the use of geometry to tell the story and engage the viewer, as well as for symmetry. We will also discuss the positive effects of mixed classes of Communications and Education majors.
4:40 - 4:55 PM	Reuse, Recycle, Re-Ceva Martha Waggoner, Simpson College Since it was published over 300 years ago, Ceva's Theorem has been used, proven, generalized and extended many times. The simplicity of this theorem about triangles is an avenue for conjecture and proof for geometry students. In this talk, I will discuss how students used Geometer's Sketchpad to explore the theorem, give a glimpse into a versatile and intuitive method of proof, and outline the possibility of extending Ceva's Theorem to other figures such as higher order polygons or polygrams. This includes a brief discussion of the connection between Ceva's Theorem and the theorems of Menelaus and Hoehn. I will also provide a list of resources for future study.

• Session 2, Saturday, August 7, 2010, 1:00 - 4:15 PM

1:00 - 1:15 PM	Centers of Triangles (for GSP 5 and Geogebra) Jane Cushman, Buffalo State College The activities used have students construct the four centers of triangles and ultimately animate the Euler Segment. There will be a GSP 5 version and a Geogebra version of the handouts. An exit ticket is also used to determine if students understand locus of points that create the circumcenter, incenter and centroid. Centers of Triangles Centers of Triangles - GeoGeBra
1:20 - 1:35 PM	GeoGebra and the Fermat-Torricelli Point Marc Renault, Shippensburg University I will discuss my positive experience using the free software program GeoGebra in geometry class. In particular, I will demonstrate a successful group project based on GeoGebra experimentation and the construction of the Fermat-Torricelli point of a triangle. (The Fermat-Torricelli point is the point in a triangle for which the sum of the distances to the three vertices is minimal.)
1:40 - 1:55 PM	Minkowski Geometry And Special Relativity Theodore Theodosopoulos, Saint Ann's School We describe an advanced high school seminar which introduces non-Euclidean geometries through an exploration of relativity from a geometric perspective. We begin with a graphical representation of the coordinate transformations between inertial frames. All along, we model space-time as a two-real-dimensional slice of C^2, with one (imaginary) spatial dimension and one (real) temporal dimension. We develop the two dimensional Poincare group of transformations that reconciles the principle of relativity with the inertial invariance of the speed of light in vacuum. This gives rise to the Lorentzian metric, together with its hyperbolic "circles." The resulting metric space is used as a laboratory for testing familiar geometric objects, like congruent triangles, right angles and areas in an unfamiliar setting. This environment also offers a concrete representation of more abstract concepts like curvature and homotopy.
2:00 - 2:15 PM	Baserunner's Optimal Path Frank Morgan, Williams College The fastest path around the baseball diamond, from a student talk to Rob Neyer's Monday Mendozas to the Mathematical Intelligencer. Baserunner's Optimal Path
2:20 - 2:35 PM	A Modern Geometry Class Works Overtime Premalatha Junius, MansfieldUniversity What infuses a Modern Geometry class with curiosity and persistence so that students are deep in conversation even after class is over? The adidactical situation will be discussed
2:40 - 2:55 PM	On Drawing Stuff Mark Schwartz, Ohio Wesleyan University Do you think it's easy to draw a straight line? It's not! Try it (no rulers* allowed) and you'll see. What about other stuff like a perfectly shaped ellipse or parabola? They're even harder. Early practitioners in geometry devised ingenious methods to draw provably correct shapes. In this talk, we consider various examples and learn why they work. * A ruler isn't allowed but it would be permissible to use a linkage. What's a linkage? That's part of the talk.
3:00 - 3:15 PM	A Golden Graph Sam Northshield, SUNY Plattsburgh We give an example of a simply constructed infinite graph that exhibits many interesting connections with the golden ratio and Fibonacci numbers. Further, this graph can be embedded in the hyperbolic half-plane so as to form a tessellation by congruent squares and such that the vertices in the graph form a quasicrystal.
3:20 - 3:35 PM	Origami and Symmetric Colorings of the Platonic Solids Lisa Mantini, Oklahoma State University The rotational symmetry groups of the Platonic solids are isomorphic to groups of permutations, but these isomorphisms are not so easy to visualize, particularly for the identification of the icosahedral group with A5. We illustrate rotation-invariant partitions of the collections of edges, faces, or vertices of a given solid through colored origami constructions and show how these colorings allow one to identify a given rotation with the corresponding permutation. Additional colorings give realizations of all of the irreducible representations of each rotation group.
3:40 - 3:55 PM	Using Biology to Teach Geometry: Protein Structure Tessellations in Matlab Majid Masso, George Mason University The three dimensional (3D) atomic coordinates for thousands of solved protein structures, freely available at the Protein Data Bank (http://www.pdb.org), generate stunning visual depictions and provide a wealth of data with which to introduce and apply concepts from geometry. In particular, we begin with a coarse-grained protein structure representation, obtained by abstracting to a point each of the constituent amino acid building blocks. Delaunay tessellation of this point set in 3D yields a convex hull of non-overlapping, irregular tetrahedra with the points serving as tetrahedral vertices; hence, the methodology provides an objective way with which to identify quadruplets of nearest-neighbor amino acids in the folded protein structure. The tetrahedral simplices of a tessellation can be categorized into five types based on whether the four amino acids represented at the vertices are consecutive or distant from one another in the primary (linear) amino acid sequence of the protein. Formulas for calculating the tetrahedrality and volume of a tetrahedron are introduced and used to identify differences in the frequency distributions of these values across the five classes of simplices, based on the collection of tetrahedra obtained from the tessellations of over 100 diverse protein structures. Matlab codes, interspersed with detailed comments, are provided for performing and graphically visualizing Delaunay tessellations, classifying the tetrahedra into five subsets by type, and calculating tetrahedrality and volume.
4:00 - 4:15 PM	Nearest Neighbors: Mathematics and Geography Leon Hannah Tabak, Cornell College Which addresses are closest to which post offices? The construction of a Voronoi diagram solves this (and similar) problems. Fortune's Algorithm efficiently draws a Voronoi diagram by sweeping a horizontal line from the top of a map to the bottom, pausing the sweep where enough information becomes available to allow the addition of vertices and line segments. A presentation of Fortune's Algorithm has helped students enrolled in a study of Geographic Information Systems to understand how a function of the software that they use works and to appreciate more fully the role of mathematics in answering geographic questions that arise in fields as diverse as biology, geology, business, and politics. Interactive, animated computer graphics show each step in the construction and inform the discussion of the character of this problem and the principles applied in its solution.

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

This page was last modified on Tuesday, July 27, 2010.