I run an after-school math club for fourth and fifth graders to share ideas and puzzles with a strong mathematics content without necessarily appearing so. It is not a competition-oriented group. Instead, I offer differentiated challenges, encouraging exploration and, I hope, some joy and inspiration.

Looking for another activity for the club, I found the NCTM Illuminations Pinwheel activity. Here’s an easy-to-follow YouTube™ video  for those who might benefit from a video demonstration. I liked the visual-spatial, geometric, and fine-motor skills that the activity would require for these younger students. Following are two examples that my fifth-grade daughter made during the club activity. The pinwheel is an eight-piece modular assembly. Modular origami creates shapes using multiple pieces of similarly folded paper that are then assembled.

To expand on that enthusiasm, I looked for other modular origami activities that the students could approach. One of our language teachers suggested that we try a three-dimensional origami star. (See the stellated icosahedron in the table below.) It was a complicated, 30-module construction, but the students were highly motivated. Several videos described this shape; we used https://www.youtube.com/watch?v=R1XTt73diyY&feature=fvwrel. Watching students persevere through the folds and assembly was deeply inspiring and motivating to me.

One of the more fascinating parts of this activity happened during module assembly. To put the star together, we used three pieces of paper folded into one another to create each equilateral triangular pyramidal “spike” (see video).

• We noticed that at each vertex of the triangular bases, the lateral edges of five pyramids met. One of these 12 vertices can be seen at the top of the image in row 1 of the table. Counting the outward pyramids in the completed figure, we found 20. That’s the number of underlying triangular bases.
• The shape used 30 pieces of paper with each folded in half forming parts of two different pyramids, making 30 base edges.
• From Euler’s formula, I recognized the formal underlying structure. The overall star was really a stellated icosahedron—an icosahedron with a pyramid on each face. And if we could stellate an icosahedron, why not the other Platonic solids, too?

Using the same folded modules needed to construct our first three-dimensional star (shown in the YouTube video), we went beyond our pictures and instructions to create shapes that mathematics told us were there but that none of us had ever seen before. The following table shows our other creations—making one for each Platonic solid. (The stellated dodecahedron was the most difficult, with “loose” pyramids that required tape inside during assembly to hold together.)

It was amazing to see all these three-dimensional objects assembled from the same module. The key is in the orientation. Enjoy!

(You can also see the Platonic duals in the paper needed for each star—a task that was beyond my math club students.)

CHRIS HARROW, [email protected], a National Board Certified Teacher, is the mathematics chair at the Hawken School in Cleveland, Ohio. He blogs and presents nationally on the educational uses of technology and on Computer Algebra Systems (CAS) in precollegiate mathematics. He is also the recipient of a Presidential Award for Excellence in Mathematics and Science Teaching.