Comparing Function Types

Place a single water droplet in an empty football stadium, and add one gallon every minute. The water level rises steadily and predictably, etching a perfectly straight line on a graph of volume over time. But if the volume of water doubles every minute—one drop, then two, then four—the early minutes feel deceptively similar to your steady effort. By the time you reach the final few minutes, however, the stadium is inundated in a catastrophic roar of water. This is the profound difference between additive progression and multiplicative explosion, a mathematical distinction that governs the compound interest in a retirement account, the trajectory of a tossed ball, and the spread of a biological population.

As a mathematics teacher, your objective is not simply to teach your students how to plug numbers into formulas. Your objective is to teach them how to read the physical world through the lens of mathematical models. To do this, we must precisely categorize how quantities change.