We’re in the middle of a unit on factoring, and I keep thinking about how rare the factorable polynomial really must be in the wild, and how weird it is that we give them so much time and attention. Students must feel like the unfactorable polynomial is the rare find, when isn’t it really the other way around?
Sometimes I feel like we’re looking at math through a telescope, backwards, in the way we’re asked to spend so much time on certain topics as part of our current high school math curriculum. Who steps back and opens the other eye to ask “why have we focused our teaching on this? And why are we teaching it this way?”
I have gotten lots of really fantastic questions and noticeably increased engagement out of my students the past few weeks, even as the cold February days drag on and everyone is itchy for a snow day that won’t come. I don’t know whether to attribute it to the fact that we’re trekking into new territory after a 1st semester with many familiar topics from 8th grade math, to give myself some props for making better choices in planning good questions and activities, or to credit a level of comfort between me and my students that has finally been reached after 6 months together. Whatever the reason, it has been a good and exhausting few weeks of teaching.
The “Word Problem Wednesday” bellwork problem today was some dumb textbook question about hourly rate and flat fee to rent something, where the total cost and number of hours rented were given for two different customers. A pretty straightforward systems of equations problem that I was surprised to get more milage out of than I expected. Several students had a hard time setting up the equations to model the situation, I think because they are so used to being given the rate/slope and initial cost/y-intercept and varying the number of hours for input and total cost for output. They can build that kind of equation in slope-intercept form in their sleep. This problem had that flipped – the rate and initial cost were the unknowns. A nice moment happened in both 6th and 7th periods when one student in each class realized that a bit of logical thinking could get you the solution faster than the algebra; talking about this strategy after solving the system algebraically with elimination illuminated (I think) some nice connections between the context of the problem and the symbols on the page.
I’m pretty careful to speak precisely about mathematics when I’m teaching. So I was rather delighted yesterday when a student called me out after I asked if any of the trinomials we were looking at had a greatest common factor. What I meant, of course, was “do any of these trinomials have a common factor that we need to factor out as a first step?” but what I asked was “do any of these problems have a GCF?” All but one of my sweet and eager students answered back loudly, “no!” while one kid protested – “yeah they do, Ms. Kernodle. One?!” I’m pretty sure he genuinely felt he had knocked me down a peg or two, but in actuality I just needed a moment to file that great moment away for future use. [Besides, if my my mathematical ego was that fragile I’d be in a lot of trouble.]
We’ve really only seen quadratic trinomials at this point, but you can’t keep kids from wondering “what if?” So I’ve praised and highlighted (with squeals and trumpet noises) the questions like “will we only be able to factor these kind of problems if they look like that? What if that exponent up there on the x wasn’t a 2?” Yeah, “what if,” kid. Thank you for seeing a star or two through our telescope and imagining the night sky.