Monday, November 24, 2014

Mr. Whalen Goes to Chicago: A Problem in 3 Acts

By now you all know that Mr. Whalen is a talented teacher.  But what you don't know is that he is an accomplished long-distance runner!   I suggested he use his recent participation in the Chicago Marathon as the basis of a 3-act with our fifth graders, and, well, he ran with it.

ACT 1:

• What do you notice?  What do you wonder?

 There he is, in the middle of the photo, with the black headband.
The chatter started immediately:
• Is that Mr. Whalen?
• When was the race?  How long was it?
• Where did he run?
• Was he running for charity?  How long did it take him?
Rich explained that they were looking at a photo taken of him running the Chicago Marathon, and that for the next few periods they would be working on finding the answer to one of their questions.  That set the stage for...

Act 2:

 This was the question that would inspire the problem solving activity.
He asked both classes what they thought they would need to know in order to figure it out:

 What the AM class wanted to know.

 What the PM class wanted to know.
Rich and I looked over their requests.  We decided to give them most of what they wanted, and would wait to see if they could separate the essential from the non-essential.

Act 3:

 Rich divided the class into partnerships and provided each with this information sheet.
But before they got started, we wanted them to make an estimate.  In  order to give them some frame of reference, he showed them the following: a time-lapse video of the entire 26.2 mile course.

The kids sat back and watched.  As the video progressed, the kids began to make some observations and comments...

• "Why would someone want to do this?"
• "You went to Chicago?"
• "I'm getting tired.  Did you really run all that way?"
• "It's 26 miles?  I can't survive 10 miles without falling asleep!"
...and when it was over they had a much better sense of the length of the course, which they were able to translate into some quite reasonable estimates.  One student justified her thinking by explaining that her father had run a half-marathon in 1 1/2 hours, which she doubled to get an estimate of 3 hours.  Another student explained that he had run a mile in about 10 minutes, so figured Mr. Whalen's time at about 260 minutes, which he converted into 4 hours and 20 minutes.
We let them work for a while, then called for a mid-workshop interruption.  We had noticed that most groups had started in on the problem by attempting to find the elapsed time between his start and finish, but we were interested to know what information they felt was not useful.  Most students agreed that the total number of runners (40,802) was unimportant, but one student disagreed: "Maybe they all got in Mr. Whalen's way and made him run slower!"  A minor dispute erupted over the significance of his coming in 7,230th place overall. That's when I overheard one student whisper to his neighbor,
"That's bad.  But don't tell Mr. Whalen."
"No it's not," his neighbor responded.  "Actually it's good!"
We let them get back to work, and the answers began to roll in:

 Uh-oh.
Clearly they could not all be right.  The most common mistake was treating his times, 11:16:58 and 7:35:08, as whole numbers that could be subtracted following the normal regrouping rules:

 We had covered this last year.  Obviously the lesson did not stick.  Although some groups were able to convert 3:81:50 into 4:21:50, it still did not help.  Further complicating matters was the fact that all these times seemed reasonable given their estimates.  Except, that is, for one group, who somehow came up with a time of 34 hours and 14 minutes.  This elicited a comment of, "What the heck?!" and a return to the drawing board.  Rich and I were pleased that the group had used their estimate to realize their answer was unreasonable.
After about a period and a half of work, the kids began to realize that the elapsed time number line they had explored last year was the better option:

 Something like this...

 ...or this.

Understanding why operating with time does not always yield the same results as operating with whole numbers was a very difficult concept for many of the students to understand.  Even after repeated explanations, I could tell that their knowing nods and comments of, "Oh yeah, now I get it!"  were not sincere.  Rich and I decided to revisit this in the future, perhaps with an "Always, Sometimes, Never" activity, or another problem solving project that would have elapsed time embedded within.
Yet overall we were pleased with the 3-act, and there were some side benefits to the project, including:

• Two more estimation activities, one for the first place time and one for the last place time;
• Some spirited counting circle activities centered around counting by tenths from 0 to 26.2;
• An attempt by some students to find his time by multiplying his average speed by 26.2;
• A lively class discussion about whether coming in 7,230th place was good or bad.
And in case you're interested:

 Eliud Kipchoge won the Chicago Marathon with a time of 2:04:11.