At the end of last week I came up with a “guessing game” to play with students in order for them to be able to come up with their own problems for multiplying/factoring polynomials. The game has one sentence: “I am thinking of two numbers that multiply to make ___ and add to make ____.” This helped them to see the connection for the standard of multiplying/factoring polynomials. Today in one of my classes we were still working on making this connection and students asked these questions. There is no way to express how awesome it is when students ask these kinds of questions during class. Who cares that it might take your ENTIRE class off course to what you are trying to accomplish. When questions like these are being asked you know you have inquiring minds thinking about mathematics in a new way than they normally think about mathematics (boring). They are genuinely interested in this. If we can get more questions like this to come out in class we know that our students are students are starting to think mathematically!
P.S. The guessing game picture question was not written down on the board in its entirety it went something like “Could we make a guessing game where you would do the opposite of multiply and add, like maybe divide and subtract?”