Slope

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!

Twitter Math Camp

 

I’m back from Twitter Math Camp, which was one of the most amazing experiences I’ve ever had. I’ve never made friends so easily, but I was so exhausted afterwards, because we pretty much spent 96 non-stop hours together (except for sleeping time). When people asked me if I had vacation plans this summer, I mentioned I was going to St. Louis for “a conference” and was told it didn’t count as vacation. But oh, it totally did.

 

I’m not sure whether I should blog about the conference or the after conference. I’m a terrible note-taker, so I’m sure others will better be able describe what went on, but highlights include spontaneously lesson-planning with Karim of Mathalicious, community-building and website ideas with Sam Shah, Megan‘s totally awesome Interactive Notebooks talk. But honestly some of the best sessions were the My Favorites… sessions, where people just went up for a few minutes and shared something awesome they did. And it was so much awesome. I also loved how, when something great was said, everyone in the room would say “Someone tweet that.”

 
As for after-conference events, Pi Pizzeria was actually quite good (and so I appreciate a Deep Dish Pizza as being something tasty, but not pizza). The brewery tour was nice, even though is was super-hot and I don’t like beer, but as before, the company was so good. Anyone going to St. Louis (or anywhere close) needs to visit the City Museum, an amazing experience, even if Max Ray did almost lose his wallet from up high. And I was convinced to go see Magic Mike by Julie and Sam, and seeing it with them and the other tweeps (like Marsha) made it hilarious. (Julie taught the whole audience a special dance!)

 

I think the thing that sums it up the most was our final activity:

 

I’ll post about my talk and what I shared later. Have a lot of chores to do now.