I’m usually pretty good at thinking on my feet—giving impromptu explanations, coming up with good examples, and tweaking arguments for the crowd in front of me. But today in one of my seventh grade classes, as time was winding down, I found myself grasping at straws. The class had been working on a MATHCOUNTS worksheet of probability problems that included the following:

*Students at Wooden High School go to school 180 days a year. Ms. Hines, the geometry teacher, assigns homework 108 days a year. Mr. Chien, the biology teacher, assigns homework 105 days a year. On a randomly selected day, what is the probability that a student in Ms. Hines’s geometry class and in Mr. Chien’s biology class will not have homework in either class? Express your answer as a common fraction.*

We were talking through this problem together, and after writing out the relevant fractions and reducing them, we got to the point where it seemed like we should multiply them. This, based on lots of other examples of making lists and tree diagrams and such. But there was some degree of puzzlement in the room about exactly *why* we should be multiplying in this case, and I just couldn’t conjure up a useful picture or diagram or explanation.

And then the bell rang. Stumped for the day, on this problem at least. Always more to think about, and another day to share it.

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