Strange Dissection

We know that any two polygons with the same area can be "dissected" into one another - this can be done by dividing them into a bunch of subpolygons. From the picture below you can imagine how to dissect a 3x5 rectangle and reassemble it into the shape right next to it (a 4x4 square with two corners missing).
Now the challenge is to do this using the least number of pieces possible. More specifically, you should cut the rectangle in two (connected) pieces and glue them back together to get the other figure.

A note for the mathematicians: this problem has a solution, and it is not of the Banach-Tarski type - the pieces can actually be cut using just an ordinary pair of scissors.