
Lenart Sphere
Most surfaces, including our own planet Earth, are not flat. So, its important for students to realize that the familiar Euclidean geometry is not the only possible geometry. Yet most high school geometry books mention nonEuclidean geometry as a historical aside if at all. This is partly because students havent had the tools to successfully visualize and investigate geometry on surfaces other than the plane. Imagine learning planar geometry without a flat surface to draw on. Its just as hard to learn spherical geometry without drawing on a sphere. The Lenart Sphere geometry construction materials allow middle and high school students to investigate spherical geometry using a real sphere and the spherical equivalents of a compass and straightedge. With these new tools, students can compare and contrast relationships on the sphere with the corresponding relationships of Euclidean geometry. Working with nonEuclidean geometry keeps students thinking fresh and deepens their insights into planar geometry. Investigating spherical geometry requires students to think creatively. And handson work with the Lenart Sphere is a great opportunity for students to increase their spatial visualization skills. Spherical geometry also has many realworld applicationsjust ask any engineer, physicist, astronomer, or global navigator. You can easily make the Lenart Sphere a regular part of your geometry program. When your students finish a geometric investigation on the plane, ask them whether the results would be the same or different on the sphere. The Lenart Sphere gives students a tool to develop and test ideas in a new geometric system. The Lenart Sphere Basic Set includes: A transparent plastic, eightinch sphere
 A ringshaped support, called a torus, to place under the sphere
 Hemispherical transparencies that fit over the sphere so that students can draw on these with colored markers and cut out shapes with scissors
 A spherical ruler with two scaled edges for drawing and measuring arcs, angles, and great circles on the sphere
 A spherical compass and center locator for drawing circles on the sphere
 A set of transparency markers for writing and drawing on the sphere and transparencies
 A hanger for displaying your spherical constructions and designs
 A 16page booklet, Getting Started on the Lenart Sphere, with activities you can begin as soon as you open the box
 A fourcolor polyconic projection of the earth that you can cut out and transform into your own globe


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