Grade Level: If you teach any kind of geometry to kids and want to do this activity, do it. The amount of scaffolding and depth of material they should be expected to digest varies with age of students but if you give them the chance, they will probably surprise you.
Materials: Giant spherical punch balloons, colored markers, papier-mâché materials
Prerequisite: Exposure to Euclid’s parallel postulate, reminder that the sum of the angles of a triangle is 180 degrees, some ideas about distance.
Driving Question: How can we rethink what we know about geometry, and use it to create the next World Cup soccer ball?
Final Product: Papier-mâché sphere with artistic renderings of surprising results of spherical geometry for future World Cup soccer ball design.
Step 1: Have students get into groups of two or three. Give each group a balloon. Ask them to get their smart phones out and look up the definitions of lines and triangles in Spherical Geometry and then draw some on their balloon with notes about the definitions. Then have them look up theorems in spherical geometry and draw diagrams of those theorems on their balloon. You can decide how many of these theorems you want on there and which ones should be specifically included. Have them write down the corresponding theorems of Euclidean Geometry on paper. Make a class poster of the theorems and the corresponding Euclidean ones. Each group should contribute to the poster.
Step 2: Give each group another balloon or have them use their original balloon, if for some reason they haven’t popped it yet. Have them papier-mâché the balloon. Let it dry. While they are doing this, they should be discussing how to make a design for the next World Cup soccer ball using results from spherical geometry (they should preview old designs as examples).
Step 3: Draw, color, paint designs on their finished spheres. Display spheres around the room.
Assessment: Talk to your students as they are working. You will know who is learning and what. Ask them questions to get them to think more deeply. Make jokes with them. If you need to write stuff down, do it, as long as it doesn’t feel like a test.