Have you looked at Tomoko Fuse's book _Unit Origami_? It describes ways of making geometric figures from several sheets of origami paper folded similarly. Unfortunately, these figures often fall apart when I try to make them.
_Patty Paper Geometry_ by Michael Serra is a book describing geometry projects for students your age. The idea there is to fold pieces of waxed paper and observe the geometric figures formed by the creases. Can you construct a square by just folding paper and marking points on it?
Similar ideas are presented in the book _Geometric Exercises in Paper Folding_, by T.S. Row. The instructions in this book seemed a little difficult to follow, though.
Heidi B. firstname.lastname@example.org University of Washington