Given a square piece of paper, describe how you can cut a portion out of the paper and reattach it in a different location to produce an irregular polygon that tessellates

Cut from 1/3 of the way along one side to 2/3 of the way along the opposite side. Joining the pieces at either of the matching sets of short sides produces a tessellating shape. 

Joining the 2S/3 sides produces a pentagon with a base of length S and two sides of length S/3 perpendicular to it. 

Joining the S/3 sides produces a pentagon, also, but this one is concave. It still tessellates.