Engaging Undergraduates in Geometry Courses Session #1  Friday, August 3, 2012, 1 PM  3:15 PM Session #2  Saturday, August 4, 2012, 1 PM  2:55 PM Organizers:
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 nonEuclidean geometries and provide opportunities for comparisons and contrasts between the two.
In
this session, we invite presentations that address the following questions:
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 #1  Friday, August 3, 2012, 1 PM  3:15 PM, Hall of Ideas F
1 PM  1:15 PM 
The Pizza Theorem and the Joy of Discovery Michael
Nathanson, Saint Mary's College of California The best mathematics course I took as an undergraduate was Tom
Banchoff’s studentdriven class in geometry. This
course began with a list of ten challenging questions and evolved organically
based on student efforts at solution. This experience was my first
opportunity to explore and research mathematics and had a profound impact on
me both as a student and as a teacher. It also introduced me to one of my
favorite geometry problems, the Pizza Theorem, which was recently written up
in Mathematics Magazine. I will demonstrate this theorem and its
generalizations; and discuss how I have used problems like this to recreate
Professor Banchoff’s active, exciting classroom
culture which I enjoyed as an undergraduate. 
1:20 PM  1:35 PM 
Two Geometry Problems Aaron
Hill, University of North Texas A geometry teacher might ask: “How can I help my students to develop
the visualization skills (or the reasoning skills) that are important for
studying/exploring/applying geometry?”
We’ll discuss two important aspects of an answer to the above
question: Rich mathematical problems and substantive student engagement. Then we’ll discuss two geometry problems
that are simple to state and naturally interesting (increasing the likelihood
that students would be substantively engaged) and that require important
visualization and reasoning skills (so in some sense they are mathematically
rich). 
1:40 PM  1:55 PM 
Elementary and Advanced Coordinate Geometry Exercises on a
Single Triangle, with Euclidean Connections J
Bradford Burkman, Louisiana School In my teaching, I make extensive use of triangles in the
plane, and connect the techniques back to Euclidean geometry. Using a single
triangle for several exercises shows students the intricate symmetries and
depth in a simple figure, and the beauty of the
resulting diagrams can encourage students to continue to explore. In
lowerlevel classes I use the centers along the Euler line, with the
associated lines, concurrences, collinearity,
circles, and distances, as introductory practice and as a culminating course
project. In higher classes I use the segments that cut the area of a triangle
in half as an occasion for students to practice parameterization, limits,
trigonometry, and conic sections [yes, there are hyperbolas].
Technologically, my students use GeoGebra to
explore, Sage to do the heavy symbolic computations, and TikZ
to make beautiful diagrams. We will
look at six ways to find the area of a triangle, the Euclidean underpinnings
of the formulas for slope, midpoints, distance, and equations of lines, and
the rich mathematics we find when we cut a triangle in half. We will look at
the technology, and explore the qualities of “good” exercises and how to find
them. 
2:20 PM  2:35 PM 
Geodesic Intuition Michael
Kerckhove, University of Richmond According to the Ribbon Test, developed as a teaching tool in
David Henderson's book Differential
Geometry: A Geometric Introduction, a curve lying on a surface in _{} is a
geodesic in that surface only if a stiff ribbon can be laid flat against the
surface with its centerline in contact with the curve. Coupling this
condition with Clairaut’s relation for geodesics on
surfaces of revolution and the important idea of local symmetry along a curve
offers the opportunity for an engaging and intuitive discussion of the
behavior of geodesics on surfaces. 
2:20 PM  2:35 PM 
Developing Intuition for Hyperbolic Geometry Ruth
I Berger, Luther College I want the students in my Euclidean and nonEuclidean Geometry
course to develop a feel for hyperbolic geometry, basically replacing their
Euclidean intuition by a Hyperbolic one. I start the course by introducing
them to 2 different worlds: “Escher's
World” is as a disk populated by inhabitants in which everything shrinks
towards the outside. The “Green Jello World”,
inhabited by fish, consists of Jello that is less
dense in one direction, but infinitely dense at the end of the world.
Students realize you can get from A to B with much fewer steps/flipper
strokes by not necessarily following a Euclidean line. They naturally come up
with curved looking paths! Throughout the course, whenever we ask if a
certain fact should hold in Hyperbolic Geometry, we first investigate it by
drawing pictures in these worlds: “How many lines through P are parallel to
L? Can a line lie entirely in the interior of an angle?”.
Having this hyperbolic intuition makes it much easier for students to then
write formal proofs in hyperbolic geometry. 
2:40 PM  2:55 PM 
Finding a Balance Between Rigor and Exploration in a
NonEuclidean Geometry Course
Jeffrey
Clark, Elon University This presentation will discuss approaches to teaching
nonEuclidean geometric content to students whose prior exposure has only
been to Euclidean geometry. It will
discuss both a rigorous framework for the material as well as software to
support student exploration, and will be aimed at firsttime instructors. 
3 PM  3:15 PM 
Imagine This: 600 Cells in 4D
John Wasserstrass, UWRock County This
talk will illustrate how the 4D regular polytopes
can be used to challenge the geometry student to think outside the box. I
will start by showing how the rectangular coordinates for the tetrahedron,
octahedron and cube can be extended into all higher dimensions. Students can
be challenged to show that Euler's formula for polyhedra
works for these. Then I will show how Descarte’s
rule for defect can be used to show that besides these 3, there can be only 2
more 3D and 3 more 4D regular polytopes. Next, by
aligning the icosahedron and dodecahedron with the cube, we will obtain the
coordinates for them in terms of the golden ratio, using the golden brick
with diagonal 2. Models will be used to show the construction of the 120 and
600 cell 4D polytopes, which are duals of one
another. Finally, the coordinates of the layers of vertices in the 600 cell
will be derived from the appropriate 3D regular polyhedron 
Session #2  Saturday, August 4, 2012, 1 PM  2:55 PM, Hall of Ideas F
1:00
PM – 1:15 PM 
Are We There Yet? Distance and
Persistence in the Poincaré Model Jason
Douma, University of Sioux Falls Mathematics
knows no shortage of existence theorems. One benefit of studying geometry is
the opportunity it provides for realizing or visualizing the objects whose existence
is assured in its theorems. These opportunities come in the form of compass
and straightedge constructions, software visualizations, and in some cases
through direct analytic calculation. Unfortunately, a onesemester course in
geometry offers precious little time to develop the scaffolding necessary for
fullfledged analytic calculations in the context of hyperbolic geometry.
This talk will examine a challenging exercise from an upper division geometry
course which draws on students' knowledge of Euclidean analytic geometry to
locate coordinates of points specified by distances in the Poincaré model of hyperbolic geometry. In addition to
helping students better grasp key distinctions
between Euclidean and nonEuclidean geometry, the exercise has inspired
students to persevere in solving novel problems. 
1:20
PM – 1:35 PM 
Hubcap Geometry David
Eugene Ewing, Missouri MAA What
geometry can exist on the surface of a Hubcap, Volcano, Saddle or a Donut?
Teach Foundations of Geometry more effectively by having students create
their own geometries on these surfaces. Several lessons will be demonstrated,
including Creating Definitions, Exploring Shapes & Their Relationships,
Formulating Theories, and Writing Proofs. 
1:40 PM – 1:55 PM 
Propelling Students into the
Projective Plane Sam
Vandervelde, St. Lawrence University The
concept of the projective plane might come across to students as either
arbitrary, unnecessarily complicated, or both. However, there are a variety
of ways to naturally motivate both the construction of and the utility of
this elegant geometry. In this talk I will share a collection of inclass
activities, puzzles, and results that have proven to be effective in helping
students to make the transition from the Euclidean plane to the projective
plane. 
2:00 PM – 2:15 PM 
Comparison of Quadrilateral Definitions
in Euclidean and Non Euclidean Geometries Filiz Dogru, Grand Valley State University; David
Schlueter, Vanderbilt University; Jiyeon Suh, Grand Valley
State University The
students are familiar with the quadrilaterals definitions before starting the
Geometry course. While we analyze properties and seek deeper understanding of
the definitions of quadrilaterals in Euclidean Geometry, we investigate which
one of those definitions works in the non Euclidean geometry (hyperbolic and
elliptic). Through class activities students are discovering the answers. In
this talk we shall share some of the students' examples and discuss
difficulties that they have encountered. Additionally, we shall talk about
9point circle in Euclidean and Minkowski
geometries. 
2:20 PM – 2:35 PM 
Teaching Mathematical Maturity through Axiomatic Geometry
Brian Katz, Augustana College Mathematical maturity includes the skills to communicate with
precision, attend to detail, and interpret results through the epistemologies
of the discipline. I will describe an inquirybased Geometry course
structured around an axiomatic development of Euclidean and Hyperbolic
Geometry, and I will analyze student products for evidence of changes in the
level of mathematical maturity. The evidence will include a comparison of
concept maps about mathematical truth from before and after the course as
well as student reflection writings about the axiomatic method and their own
development in proof construction and communication.

2:40 PM – 2:55 PM 
Topics in Spherical geometry for
Undergraduates Marshall
A Whittlesey, California State University San
Marcos A
century ago, spherical geometry was a standard part of the mathematics
curriculum in high schools and colleges. Today most mathematicians only learn
about it as a short topic in geometry survey courses. In this talk we explore
the idea of teaching spherical geometry at greater depth by discussing some
key theorems of spherical geometry, short proofs and applications to other
areas such astronomy, crystallography, and polyhedra.
We discuss how to use different techniques (synthetic versus analytic) to
advantage in this subject in the hope that the student will benefit from
thinking about when each method is appropriate. We also think that comparison
of theorems of spherical geometry to those of plane geometry are a good way
for the student to see in a tangible way how changing axioms results in
different theorems. We think that exposure to spherical geometry is
particularly good for future high school teachers, but also that more
mathematicians should be aware of its theorems and applications. 
This
page was created and is maintained by S. L. Mabrouk, Framingham State University.
This
page was last modified on Friday, May 31, 2012.