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.”

(c) Mathalicious 2011

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!

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Comments on:"Slope" (6)I Speak Mathsaid:I love this! Thanks for the update of our fascinating TMC discussions. Also, I am loving the wackiest stairs! How fun is that? :)

James Clevelandsaid:It’s pretty great. I love the kids struggling with the different measurements, trying to figure out how to relate different stairs when I used a pen for one and a cell phone for another. This year I decided to make it more student-centered, so I’ll probably write that up when I’m done.

Karimsaid:Love it. Even if we think of slope geometrically, though, does that really imply no units? After all, if we ask students, “Which staircase is steepest,” they’ll have to measure something to determine the answer. But what, exactly, are they measuring? If they say, “This staircase rises 3 inches for every inch that it runs,” then the units are inches. “7 units for every 2 units,” units. On the other hand, if a kid says, “The staircase rises 9 for every 4,” won’t we as teachers naturally say, “Four *what*?” If so, then does that mean that we naturally include units, even when we don’t necessarily care what they are?

James Clevelandsaid:Well, think of this. I have one staircase that I measured, and its slope is is 1 pen up, 2.5 pens over. Then I have another staircase and its slope is 1 iPad up, 2 iPads over. But how could I possibly compare those two staircases, when I used such different units? I don’t even know the conversion rate for iPad to pen.

But we can compare them if we talk about the pure ratio. The ratio of the height and depth of the first staircase is 2/5, and the ratio of the height/depth of the second is 1/2. And those ratios are directly comparable.

That abstraction is one of those key elements of mathematics. It’s like when we compare 6 oranges and 6 airplanes. They are obviously not the same. But their amount is the same. That’s something that’s equal in the two.

GregTsaid:Well, and when I teach it for the first time, I do tend to make the distinction, using “unit rate” when discussing scatterplots and then after shifting into the geometry unit, using “slope” as I make the equivalence. Of course, with the internet and everything out there, after I come out with unit rate I sometimes have students saying “isn’t that slope?” (“Depends. What’s slope?” “I don’t know I heard it was something like that in this course.”)

Of course, it could always be more confusing… Tangent is effectively another word meaning slope (assuming you orient the triangle correctly). With a “run” of 1, you’ve now got the connection back to the angle. (An argument for another course…)

Steepest Stairs Redux « The Roots of the Equationsaid:[…] an idea from Dan Meyer. It took about a period, and was mostly teacher-led. But after arguments and deep thinking about slope, I wanted to go into the lesson deeper, so I turned it into a […]