At Twitter Math Camp, Karim Kai Ani and I debated for a bit on what slope really means, and how best to teach it. Since slope is the upcoming topic for this week, I thought it would be good to reflect back on our arguments.
Karim argued that slope should always have units, and that removing the units created a contextless concept that made it difficult for students to grasp. I argued that, while that is true and units are useful in many cases, the concept of slope as a unitless ratio is an important concept, digging deep into what it means to be a ratio, so that a line with a slope of 2 could be 2 miles up, 1 mile over, or 2 cm up, 1 cm over, it didn’t matter. The differences are exemplified in two of our lessons: my “Steepest Stairs and Wacky Measurements” (soon to be updated) and his “iCost.”
I mentioned this debate at dinner last night to my boyfriend, who is a math PhD candidate. He said what we were talking about reminded him of the difference between a rate and a ratio. He said that a ratio was a “quotient of quantities of the same unit” and a rate was a “quotient of quantities of differing units.” Further clarification was that a ratio’s units had to be the same dimension, while a rates did not.
So then, really, the question becomes, is slope a rate or a ratio?
It’s both. Karim argued for rate but that’s really just the algebraic or calculus-based definition of slope. My argument for ratio was a geometric one. Both are important, and are related, which is why they go by the same name.
But I wonder if it would be easier if the concepts had a different word. What if we only used “grade” or “gradient” for the geometric definition, and slope for the algebraic one? Or slope for the geometric, and just rate for the algebraic? The problem is they are so intertwined. For which there is only one person to blame.
Damn you Descartes!