Flatland and sphereland. There are many aspects of spherical geometry that could be investigated. Explore congruences of triangles on a sphere. One can also make very effective models with plastic spheres from a craft shop and cut-off plastic containers for rulers. Explore quadrilaterals and their symmetries on a sphere.
Build models showing that parallelograms with the same base and height have the same areas.
This can lead to a purely visual proof of the Pythagorean theorem, using a physical model based on dissections. Consider tiling the plane using shapes of the same size. In particular it can be shown that any 4-sided shape can tile the plane. Check the Martin Gardner books. List them all.