One day I was reading a board book to the kids about different tools, and trying to tell them about a jigsaw. Suddenly I decided the best (although perhaps not the quickest) way to teach them this was to make our own jigsaw.
I got a scrap piece of MDF, and we drew our pictures on in pencil. We each painted our own objects in the picture, and when the paint was dry, outlined the shapes with a black marker pen.
We took the picture outside and I cut it up with a jigsaw (while the kids mostly ignored me, except for occasionally asking me if it was ready yet). When it was all done, and dusted off, we got do do our own puzzle!