Lesson Exploder: Toll Highways

Part 1:  The Hook

I was reading Dan Meyer’s blog the other day, and in the same post about ideas that had been bouncing around in his head, found myself suffering from a mindworm (maybe the reading version of an earworm?).

The idea of creating space where the messy creation of math lessons gets visualized, and the curtain is pulled back is super appealing to me.  It’s something that I try to help my student teachers see, and something that I think could be helpful to understanding each others’ processes as we glide through the math world.

The second idea that struck me was looking at the unexpected result of driving along the New Jersey Turnpike, and learning that the tolls that you pay don’t follow a linear pattern.  Being from New England, and living north of Boston, I was curious to see if the MassPike followed the same kind of weird pattern.

So I Googled MassPike toll calculator and started playing with the tool linked from the MassDOT website.  I used the tool to calculate tolls when traveling from West Stockbridge (as far west as you can leave the MassPike), and getting off at each possible toll along the way.  And to make it pretty, I recorded the data in Desmos.  This is what I got.


Weird, right?  The reasoning kind of makes sense.  There are two flat areas as your traveling through Springfield and Worcester, so that people don’t have to pay to go one exit and stay in the city.  And there’s a spike as you head into Boston, and the tolls increase for a small distance at the end, most noticeably with a $4 toll entering the Ted Williams Tunnel.

At this point, I’m mathematically hooked, and want to find a way to get my Algebra II students as excited about this as I am.

Part Two:  Development

So I’ve got this idea, and need to figure out how to take this and place this into an algebra course.  Where does it fit?  What skills does it let students develop and play with?

To me, this looks like a piecewise function, so I think I could work it into the start of the year, where I am working through linear-ish functions.

That gives me a where.  Now to think about what types of skills I want students to develop.

I’d like students to:

(1)  use technology (desmos) to graph piecewise functions, and write equations to represent piecewise functions;

(2)  collect data and use technology (desmos) to plot points, and choose a reasonable window;

(3) describe relationships in data using mathy words;

(4) connect representations of data graphically, in tables, with words, and with equations;

(5) do most of the thinking, or co-thinking with me to figure things out.

I know that I tend to make things overly descriptive in terms of laying out procedures, so it might be nice to build something, and then find places I could give students more freedom.

I want students to have access to several data sources, so I need to figure out what some different toll roads that I know of.  I’ve got two so far: MassPike, and the New Jersey Turnpike.  I know of a couple other toll roads, the New York Thruway, and there’s one around DC, but I remember hearing on NPR that that’s based on on-demand pricing, so maybe a few levels above the challenge I’m looking for.  Thankfully, wikipedia came in for the save with this list of all toll roads in the US.  Do I want to expand outside of this region?  I wonder if other toll roads behave like I think they should, or if they are all as unexpected as the NJ and MA data.  I’m also curious to see if intermediate data follows the same pattern (middle to middle points), but that might be another pickle altogether.

Here’s what I come up with as a first draft.  I’m letting myself be more teacher-centered and less student-thinking than I like here.






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