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 Iâ€™ve done and think Iâ€™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 â€”Â 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 â€˜fake the dice resultsâ€™ and have students determine whether or not there IS a solution to Kalebâ€™s problem; can they prove they have exhausted all possibilities?