Kaleb Allinson, with a great end-of-class factoring exercise:
I tell my students that my dice add to 19 and multiply to 88 and ask them to guess my dice. I try to play this at the end of class for a week or two as I have time leading up to factoring. Then when they discover how to factor, this dice guessing skill is very helpful. They always realize what I’ve done and think I’m really tricky.
There are other ways to do this, of course, but the dice randomize the factors and I think that’s important. It says to the student, “Whatever algorithm you’re rolling around in your head right now — it’ll work for any whole numbers. The teacher isn’t putting her thumb on the scale. She’s giving you numbers she can’t control.”
Another extension is to ‘fake the dice results’ and have students determine whether or not there IS a solution to Kaleb’s problem; can they prove they have exhausted all possibilities?