This summer, I have been training for the Prairie State Marathon on October 6th of this year. It's my first-ever marathon, and I'm totally psyched for it. In fact, I'm so psyched about running that I've already registered for the F^3 Lake Half Marathon -- which takes place along Lake Michigan

*in freaking January*-- and am planning to run the Wisconsin Marathon next May. Needless to say, I'm addicted to running beyond all measure of common sense (which I'm not certain is actually measurable).

My fanaticism for endurance running aside, I was feeling particularly inspired after reading this post by Nat Banting (@NatBanting) about a problem he's developing to determine how to minimize the amount of water wasted by his sprinkler. I found myself wondering what real-life situations I could use to similarly create an authentic PrBL experience for my students.

**The Scenario**

As I was out for a 7-mile run this morning, I realized that such an experience might lie in this:

This is a Nathan Trail Mix 4 hydration belt, which I take with me on my long-distance training runs. Each bottle has a capacity of 10 ounces, which means I can take 40 ounces of liquid with me. Most of the time, I consume Gatorade while running. With longer distances, however, I also take packets of GU energy gel with me for supplementing my glycogen stores.

(Incidentally, Chocolate Outrage is my favorite flavor.
Also, "GU" is pronounced "
goo.") |

My longest training run to this point has been 16.5 miles; thus far, I have been able to ration my Gatorade and water appropriately to get me through each run. In addition, there will be hydration stations spread throughout the course stocked with water and sports drink. However, as race day approaches, I've been frequently asking myself this question:

*Will I be able to take enough Gatorade and water with me to get me through 26.2 miles before I run out of both?*

**The Task**

This leads to the driving question I would put to my students for this task:

*What is the best plan to keep Mr. B hydrated and energized during the marathon?*

I am not yet sure what the entry event will look like, but my intention is for it to include only a few pieces of information:

- The task is to create a "consumption schedule" that tells Mr. B when to drink liquid or eat a gel packet.
- A marathon is 26.2 miles long.
- Mr. B has a hydration belt for carrying water, Gatorade, and energy gel.
- Energy gel
*must*be taken with water, and*must not*be taken with Gatorade. - There are also hydration stations located throughout the course that carry water and sports drink.

**Potential Student Need-To-Knows**

My hope is that the limited information from the entry event leads to several student-generated need-to-knows. Here are a few that I was able to come up with on my own:

- How much liquid does each bottle hold?
- How much Gatorade and how much water should be taken?
- How often should Mr. B consume Gatorade?
- How many hydration stations are there on the course?
- Where are the hydration stations?
- How many gels will Mr. B consume?
- How fast does Mr. B run?

Some of these need-to-knows can be answered rather quickly. As I mentioned above, each bottle on my hydration belt has a capacity of 10 ounces, for a total capacity of 40 ounces. If the students ask, I can simply tell them this one.

If students inquire about the number/locations of hydration stations on the marathon course, I will be able to furnish them with this map from the race web site. The map marks all of the hydration stations throughout the course.

The frequency and amount of Gatorade consumption is where I'm sure many groups will diverge in their solution paths. Some runners consume a few ounces of Gatorade every few miles or so. My personal preference is to sip about an ounce or two or Gatorade at every mile marker, though for the purpose of this task I'm not married to that notion. Chugging an entire 10-ounce bottle at any point, however, would be inadvisable, as it would probably result in me throwing up (eww).

In any case, the Gatorade consumption can be modeled with, say, a linear inequality. Is the total amount of Gatorade consumed going to be equal or less than the amount of Gatorade available to me during the race? Each group's plan will need to address this question, and there are a number of ways to answer it.

Now, I haven't forgotten about those last two need-to-knows:

- How many gels will Mr. B consume?
- How fast does Mr. B run?

These two questions are very closed tied to each other; plus, the gel consumption will also dictate the water consumption.

Gel needs to be consumed at regular intervals throughout the race in order for me to maintain my energy stores; in fact, most packages of gel carry the advice of consuming one packet every 45 minutes. With that in mind, it becomes incredibly important to know how quickly I can be expected to finish the race. Without a sense of how fast I run, students will be unable to determine how much gel and water I will need to consume.

I happen to have recorded the times of each of my long-distance training runs from the past few months. Mostly this is due to the fact that I am a shameless braggart:

Since I've recorded all of my times, however, it means that instead of giving my students my own estimate of how fast I run, I can provide them with a table of data and have

*them*estimate how fast I can run. Linear regression models, anyone?

**Where To Go From Here?**

I'm only about 8 hours removed from when the idea for this task first formed in my head, so naturally it's nowhere near perfect. There are many pieces of the task that I was admittedly rather vague in articulating. But, I do think I'm onto something cool here.

One thing I am not sure about is how to have my students present their solution. Live presentation? Blog post? Physical document? Scribbles on the back of a napkin? All of these options and more?

Any thoughts? This is the first PrBL idea I've really come up with on my own, so I gladly welcome feedback!

Awesome!

ReplyDeleteNTK - "What is the temperature during the race and how will that affect your hydration rate?"

For the final product I think you should have the kids run 26.2 miles and test out their hydration hypothesis! :)

Haha, I think should, Nate. :)

DeleteThat's also a great potential NTK -- if the students bring it up, they'll be adding to the complexity of the task!

Another twist I thought of: Suppose one or two of the bottles have a tendency to leak while I'm running? That throws another wrench into their plans.

I like it, and it could be a good way to discuss the practical vs. theoretical when using a mathematical model. It's easy to understand why you wouldn't run "negative" miles; on the flip side, if you extended the race, when would the model fall apart? (You'd have to stop and eat/sleep at some point, right?).

ReplyDeleteAs far as presentation of the final solution, I think the back of an envelope works better than a napkin. (Actually, as long as the present it somehow - and tune it along the way - I find that it usually goes well, no?).

Thanks Jeff! Great points. I hadn't thought about discussing the "practical vs. theoretical" aspect of the task, but I think it would be great to bring that up. "Real-life" math tends to be far messier than "classroom" math!

DeleteI like the race extension idea. Part of your entry doc should be showing a clip from Forrest Gump where he is running across the country for no reason. Throwing another wrench in there you could have them figure out approximately how long it would take for you to run across the country (with stops for sleep and such), how often you would have to refill your hydration belt, how many gel packs you would have to bring, and they should also be required to create a regression model for how much you will progressively miss your wife while you are gone. Maybe leave out the last idea :) Then once they crunch all that data you can have them compare and contrast their original model. They can write hypotheses about why the original model does not work long term. Just a few ideas!

Delete