Are you looking for some hands on tasks to do in your upper division geometry or abstract algebra class? If so you should check out Geometric Origami by Robert Geretschläger. This book connects the art of paper folding to geometry and algebra. Some specific topics discussed are the connections between paper folding and the golden section, the cubic equation and straight-edge and compass constructions. Rigorous proofs appear mostly in the first section of the text, and could be used as nice examples for students learning to read and write proofs.
This book will not teach you how to fold a swan from a piece of paper, but it will teach you how to fold a piece of paper so that the resulting creases form a regular n-gon and then the mathematics behind those creases is explained. The folding instructions are fairly easy to follow and the book is straightforward to read. It could also be used as a springboard or a reference for some undergraduate research.
Suzanne Caulk is an assistant professor of mathematics at Regis University in Denver, CO. She is very interested in modular forms. You can email her at firstname.lastname@example.org.