Physical models of non-Euclidean geometries are of great importance in helping students to see the need to study such geometries, and to understand the interplay of length, angle, and area in those geometries. Spheres are everywhere, so motivating the need for spherical geometry is routine. Models like the Lenart Sphere enable students to do geometric constructions on spherical surfaces. They can then compare such constructions to similar ones done on a flat surface.
The author makes clear acrylic saddle surfaces, and uses them in the teaching of geometry. At the beginning of the course, students come up a name for the surface, and this discussion always leads to reasons for studying such surfaces. Later in the course, they experiment with it to discover facts of non-Euclidean geometries. Finally, they compare constructions on it with ones on spheres and flat surfaces.
In this paper, the author discusses building the saddle surface, its connection with hyperbolic geometry, and how students can use it to enhance exploration of geometric ideas typically discussed in college geometry. Labs on comparison, metric properties, and geometric constructions are presented, including construction of Saccheri quadrilaterals and Bolyai's construction of parallel lines.