Modeling is in the air. Thursday a colleague told me he was looking at Modeling Instruction for next year. Then yesterday, Shawn committed. So I started reading up. I was a full-time modeler before teaching, so I’m biased, but I love it. In Modeling Instruction physics, students organize their year by making and testing hypotheses about observable aspects of a few archetypical physical systems. The hypotheses arise from curiosity: we see the pendulum moving, what are things we could measure? What predicts their values? Here is the world, you are already fluent in it, go make sense of it. What the students construct in response is a solid understanding of the core content of introductory physics.
I could make Algebra 2 into a modeling course. But modeling is using math to describe what you are studying; I would like to be studying the math. I’m looking for a destination such that the study of each of these functions is a step towards something pretty deep and beautiful.
These functions are the structure of everything. Yes, you can model radioactive decay with an exponential. And yes, you can model compound interest with an exponential. But the exponential, the idea of the exponential, is the abstraction of what interest and radioactivity have in common: what you get is proportional to what you have. Yes, the height of a ball over time is quadratic. Yes, as you fence off your rectangular llama pasture, making it squarer and squarer, the area is quadratic. The parabola is the abstraction of what constant acceleration and constrained rectangular area have in common: the unceasing influence of the second difference, patiently turning things its way one bit at a time, until the system is inevitably flying in its direction.
These objects, the stuff of Algebra 2, are too fascinating and too pervasive not to lead to some summit worth attempting.