Wednesday, March 19, 2014

Factors, Primes, Composites, and Some NBA Action

Love him or hate him, most agree that Carmelo Anthony is the MVP of the Knicks.  Unless, that is, you're a fifth grader in Rich Whalen's class, in which case...

 Are you kidding me?
Yes it's true, turns out Andrea Bargnani is the Knicks MVP!  How can this be?
The kids were getting a bit burned out on all the fraction work, and we still had a ways to go in the unit, so we decided to take a time out and circle back to something we explored in September: factors, primes, composites, and the game "Factor Captor".   At the time, Rich and I had played around with a project centered around football rosters.

 2013 New York Giants. A disappointing season.

 Questions.  A disappointing lesson.

It wasn't so great.  There were too many numbers, too many questions, and not everyone was secure with the requisite skills.  We got a little mileage out of it (mostly with the high achievers), but did not follow up.
I liked the idea of using basketball rosters.  Fewer numbers, and easier to work with.  My idea was to play off the rules for "Factor Captor" and have the kids take each player's jersey number, find its factors, identify the number as prime or composite, and then add the factors together to come up with a "score" for that player.  The player with the highest score would be the team MVP.  Then all the scores would be added, giving the team one final, total team score.  Then we'd set up a tournament where you would match up against another team, and the team with the highest total score would win and advance.

 First, we gave the kids the rosters and asked them to take a look and talk about what they noticed.   The rosters contained quite a bit of information and we made sure everyone understood what they were looking at.
 Making progress with the Bobcats roster.  Rich's student teacher, Shannon, helped design and create the record sheet.  There was still some confusion about which numbers were prime, especially with 0 and 1.
 Calculators were allowed.
 Why is it necessary to check your work for accuracy?  Because if you miss any factors or make a calculation error, the opposing team will "steal" those points!

Shannon had a great idea about how to change the game: instead of simply matching total score vs. total score, why not match player vs. player?  Team 1 would identify a player, Team 2 would match him with a player from their roster, and the team with the highest scoring player would receive the point differential.  Any missing factors (on either side) could be "stolen" and added to the team's aggregate score.  Then Team 2 would select a player, and so on.  The differentials would be added, and the team with the highest total would be the winner.  This added an element of strategy that was missing in my original conception, and had the added benefit of echoing the rules of "Factor Captor".

Opening Day finally arrived:

 Schedule for the AM class.

 Knicks vs. Heat.  Garbage time already?  Cole Aldrich makes an early appearance for the Knicks.

 Bargnani posterizes Chris Bosh!

 Knicks lose, 258-231.
Of course I had to get into the action.  I was partnering with one of my students (we were the Knicks of course!)  I knew beforehand that the Heat had a higher aggregate total than the Knicks, but was hopeful that by implementing some type of match-up strategy we might pull it out, and I openly discussed this during the game.   One of the boys running the Heat roster said that he didn't think it mattered, in any order the Knicks would lose.  I asked him to explain why, but he wasn't able to articulate a reason.
This got me thinking, and I pulled Rich aside.  We began to talk it through.  Did order matter?  If it didn't, then the outcome of every game was predetermined, and playing served no purpose other than having kids practice finding the difference between two numbers and doing some column addition.   Quickly Rich and I tried different combinations of only three players per side, and here's what we found:  The scores of each game would vary depending on the order the players were matched up, but the difference in the scores would always remain the same.
I called my supervisor and explained the problem.  That afternoon he showed up at my door with this:

 Using only three players per side, this is how he explained to me that order did not matter.  Here Team B will always win by 39 points.  That's why he's my supervisor!

I had to try this for myself.  I could understand why order didn't matter, but I was fascinated by the fact that the score would change.  I had to play it out on my whiteboard.

 Two hypothetical teams.  Team A has a higher aggregate score (15 point difference).

 Scenario 1.  Team A wins 31-16.

 Scenario 2.  Team A wins 43-28.

 Scenario 3.  Team A wins 18-3.
This was really cool.  In each instance Team A won by 15 points...but the score of each game was different.  So we were faced with a decision about how to proceed with the project: we wanted the kids to discover this for themselves, but having them play each other multiple times with a 15 man roster would take too long, and the inevitable computation errors would confound the discovery process.  After some thought, we decided to have everybody play the same 5 on 5 game with Eastern Conference vs. Western Conference All-Stars.

 We gave everyone a roster with the individual scores already calculated.  We asked each kid to find the aggregate score for each all-star team.  Then we put the question to the class: Given this information, will the West all-stars always win?  There was widespread disagreement.  Many felt, as I had, that the East could win given the right match-up circumstances.
We set them off to work.  They were given the option to work alone or to choose someone to partner with.

 They were anxious to get started.  The 5 on 5 game was easier to play and took much less time than the 15 on 15 game we had originally conceived.
 They used the sheets in different ways.   Rich copied them back to back so they could be used to play multiple games.

 As the scores came in the kids posted them on the chalkboard.  After a while, a pattern emerged and we took a time-out to discuss what was happening.  Seemed the West always won by 3 points.  Coincidence?  Could the East ever win?
 Confounding data got them all excited, and there was a mad rush to check and see if the results were accurate...

 ...which of course they weren't.  The most common error was using a jersey number instead of the player's total factor score.  So the scores had to be fixed.
They kept working like this for around 40 minutes.

 The board filled up...
Finally we had to call it quits.  Some were now convinced that the West would always win by 3 points, but there were many who just wanted to keep trying.  And that's where we left it.  Unresolved.
Here's what I learned:
• When you experiment, you never know what you might discover.  The original lesson morphed into something totally unexpected but really amazing.

Here's why I think it was successful:
• It wasn't really complicated.  They could all do it, and they were all in it working together.  No one person could have played out all the possible combinations, but with everybody trying we could gather some good data.
• The fact that it was connected to pro hoops made it alluring to some, but there were many who had no idea who any of the players were.  I think they got excited because I was excited.  So, yeah, enthusiasm is contagious!
Here's what I'd do differently next time:
• Start the same way, with the kids finding the scores for all the players on their rosters, but have them select the top 5 to play with.  Calculate the aggregate score for just those 5.
• Play another team multiple times.  This should go quickly.  It should not take long for the kids to get suspicious about what's happening.  Kids get a chance to hypothesize: can the lower scoring team ever win?  Is the outcome predetermined?
• Then do the all-star game experiment.  This time keep track of all possible scores on chart paper.  This way they can be cut out and then organized (lowest to highest?) to look for patterns and repeats.
And we're not done with this project.  So stay tuned for some more NBA action!