### Background

A little over a year ago I created a set of Nike+ Running-themed proportional reasoning challenges for my middle school students. The idea originated as a failed attempt at a Three Act task, and eventually turned in to a somewhat-useful website (proportionplay.com). You can read more about the project here.

Even after using these problems occasionally throughout last year, and also in various teacher workshop settings, I was never satisfied with the results. Things usually drifted into uncomfortable arithmetic, and my focus from the word “Go!” was always on the proportional reasoning element.

So after mulling things over during last year’s spring semester—and trying out some of those ideas at the start of this year—I’ve settled on a new approach to the same set of problems. I’m optimistic that this new twist will make the problems much more engaging to my own students (and maybe even to some of yours as well).

### The (Old) Running Game

The old me would start the conversation with an image like this:

Then I’d ask students what they noticed about the images. There would often be a wide range of answers (“I see orange!” “Hey, the phone battery is draining!”), but eventually we’d have covered the basics: “After 10 minutes, some mystery man (or woman) had run 1.88 km. This person apparently kept running for another 10 minutes, and then for another 5. We don’t know the total distance traveled at either of those latter points.”

Next up: Find what’s missing!

This is where we’d wander off track from my primary goal (develop proportional reasoning!) and end up stuck in some tedious arithmetic (with the tediousness increasing slightly each day in the series of 20 challenges). There’s nothing wrong with having (or building) proficiency with decimal arithmetic. However, my goal here was to emphasize and develop proportional reasoning. The doubling, the halving, the combining of these pieces to make five halves of the original… Students would stare into the blinding light of ninety-four hundredths and somehow lose track of what they were after in the first place.

Good conversations often ensued, but I was after *great* conversations. Something had to change.

### The (New) Running Game

Here’s how we roll now: I still begin with an image…

And we still do the whole noticing thing (at least on the first couple of challenges). But then I invite (encourage? demand?) students to cut some corners.

- 21 minutes and 5 seconds? Nah! That’s basically just 21 minutes.
- Double that? 42 minutes. Boom! We have our first answer: “This runner probably took
*about*(hugely important word for the rest of our running game conversations) 42 minutes to run 8 km.” - Back to 4 km in 21 minutes. Half of 21? Let’s call it 10. So 2 km in
*about*10 minutes (maybe a little more). - So 42 plus 10… That’s 52 minutes.
- But maybe our runner is no Madison Bumgarner. Maybe fatigue
*actually affects*our faithful jogging friend. Let’s hedge our bet. Maybe… 54 minutes?

And then, the reveal:

And it turns out we’re wrong. Totally and completely wrong. But now the fun begins, because we can ask:

- What happened?!
- Why were we so off?
- Did our arithmetic betray us? (No. We took some liberties with rounding so the arithmetic would be easy-peasy.)
- So what gives?
- Did this guy (or gal) speed up? Or did he/she slow down? (To me, this conversation alone is worth all the screenshots I took while running with a phone in my hand.)

Maybe we made some assumptions about our runner. Maybe we need to blow up some of those assumptions so our future “off-ness” isn’t so pronounced. (And isn’t that one of the best parts of modeling? Figuring out the subtleties of the situation… Testing and either retaining or tossing out our assumptions…)

So we play again:

And this time we come to the table with some additional information. This dude *slows down* (and seriously) as the seconds tick by. Let’s keep that in mind as we think through the next round:

- 1.89 miles… That’s pretty close to 2 miles in 16 minutes
- That’s 1 mile every 8 minutes
- 24 minutes… Let’s chalk that up as 3 miles
- 32 minutes… That would be 4 miles
- What about 36 minutes? That’s another 4 minutes (half of 8!) so I’m thinking an extra 0.5 mile. All told: 4.5 miles.

Of course, that would be if our runner followed a predictable pattern. Now the fun comes in. This is where every single student in the class could end up with a unique and not-unjustifiable answer. How do you want to hedge your bet in the end? Will the runner slow down? Have we already accounted for that with our rounding up to 2 miles? (Or have we gone the wrong way?!) And so the customization of the answers begins (ideally with much arguing and hoping and anticipating).

Me? I’m thinking we should tone down the distances… 2.7 miles in 24 minutes, 4 miles in 36 minutes. So, how did I do?

Now when you throw that next image on the board, and one kid in the back slams his fist on the table in defeat and the other pumps her fist in the air in triumph… You know you’ve got them.

### The Road Ahead

The challenges are all still available at proportionplay.com. And I’ve replaced the original (and terrible) handout with this shiny new one. I think the new handout does a reasonably good job supporting my updated approach to these tasks by emphasizing the *reasoning* involved in the entire process. That—combined with my wide open invitation to estimate since the runner is almost certainly not a same-pace-all-day robot—should shift the focus from arithmetic to proportional reasoning.

### Post. Script.

The last two challenges are both borderline terrible. Or wonderful. Or both. Here’s a preview of Day 20:

### Post^2. Script.

Another favorite aspect of these problems? Whether you round or not, there are several ways to tackle each challenge: