College Algebra, reviewing the graphs of polynomial functions. Each student has a whiteboard. We started with y=(x+2)(x-3) for simplicity.
Me: What do we know about this graph?
Student(s): It has x-intercepts at -2 and 3 (or something along those lines)
Me: What else do we know?
Student(s): It’s a parabola (or some version of that)
Me: What else do we know?
Student(s): [Crickets] (or owls or frogs or some other creature that makes noises when all else is silent.)
Me: Does this parabola open up or down?
Student(s): Up. Down. no Up. no Down.
At this point I’m shocked that they do not remember the one polynomial coefficient that they all nail down in algebra 2: The a value. But I shouldn’t have been. Rather than asking “What will the sign on the x^2 term be?” I decided to approach it differently to see if I could garner some conceptual understanding.
Me: If x is a really big positive number, like a million, what kind of number will we get for y?
Student(s): A really big number.
Me: Similarly, if x is a really big negative number, like negative a million, what kind of number will we get for y?
Student(s): [After much thought and group deliberation] A really big positive number…OH, then it opens up.
This wasn’t a huge victory, but it was satisfying. Because not a single student mentioned an a value even if they were thinking it. Additionally, when we moved to cubic functions like y=(x-2)(x-3)(x+4), they used the idea of substituting really big negative and positive numbers for x to determine which way the graph was trending in each direction. We were then able to have a nice discussion about why a graph like y=(8.5-2x)(11-2x)(x) looks similar to y=(x-2)(x-3)(x+4) when the equations have so many differences.
When students learn a procedure, it’s very difficult for them to deviate from the steps in order to solidify their conceptual knowledge. I’m very glad that on this Friday, their forgetfulness of the “steps” allowed us to have a nice discussion.
I’m seeing the same with my Freshmen. They seemingly can’t remember much beyond their own last names but their ability to reason and figure things out (and patience to do so) is much better than I remember of students from the past. My theory is it’s from figuring out apps.
This makes me so happy that they are better at figuring. Kids work to figure out games til they have it. They don’t hand it back and shrug their shoulders. I think that’s valuable.
I don’t know if it opens up or down either
A quadratic is the product of linear functions. If those linear functions are traveling in opposite directions, the parabola will open in negative direction. Many avenues to explore from this perspective.