Since writing this instructable I have revised the project and no longer feel that the tutorial I published is the best way to make this calculator -- however since it had been entered into a contest, I cannot delete the instructable. Instead I will just delete the steps (except for this introduction). Apologies for the inconvenience...
In grade school kids are taught about math "triangles." 3+4=7 or 7-4=3 or 7-3=4 for example. The same principle works with multiplication and division (3x4=12 or 12/3=4 or 12/4=3) By putting each number on the angle of a triangular flash card and by hiding one of the three corners, not only are children practicing memorization, they are grasping the concept that addition and subtraction or multiplication and division are like the flip side of the same coin. They can literally see the relationship.
My mechanical calculator is based on this triangle, but by using simple mechanics the triangle formed by the frog's feet and his clasped hands can move. A single card will show the whole multiplication table, and watching it move around is a whole lot more fun than flashcards.
I should say that I did NOT come up with the concept: William Robertson, in 1916, designed a tin toy he named the "Educated Monkey." When I fist saw it a few months ago I thought it would be fun and easy to reproduce the idea on paper -- I was dead wrong.
My desk has been strewn with pins and frog body parts for weeks as I've been trying to fine-tune the template. Changing the monkey into a frog was easy and natural; the way the creature moves is much more frog-like than monkeyish. Getting everything to line up properly was a bit tricky, but manageable. I was able to improve Mr. Robertson's design by figuring out a way to display when a number is multiplied or added to itself (i.e. 5+5) He just drew a little square which represents the number, but my template shows the actual number (you can't see me right now, but I just patted myself on the back). I found tiny aluminum eyelets at a paper supply store, but during the design process, for expediency, I used pins sticking up and tried not to injure myself as I banged my desk in frustration... Because after getting everything to work perfectly, when I finally printed my frog out in color, cut the pieces for what I thought would be the last time and used the cute little eyelets --- it didn't work. The pins I had been using held the paper body parts in place while allowing them to pivot freely. The eyelets, on the other hand, held the paper together too tightly, so in a few critical places instead of pivoting the paper stuck and buckled instead. The d... thing needs to be made with a stiffer material after all.