Monday, May 23, 2016

MVPs and Fair Teams

You will not catch me claiming that problems need to be real world in order to be relevant. I would much rather think of classroom materials as either mind numbing or thought provoking. This continuum can be applied to hypothetical, practical, or genuine scenarios (a classification system neatly summarized in a chart in this article).

I see the greatest potential in scenarios that provide elegant entrance to mathematical reasoning. If it happens to contain a real world context, fantastic. Either way, it needs to be thought provoking. 

Take a look at the chart below:

If you don't follow the NBA at all, meet Steph Curry--he's kind of a big deal. He is the first unanimous MVP in the history of the league.

As I look at the table, a few questions immediately make it thought provoking. Some provide a more obvious link to a curricular outcome (in this case, systems of equations), but all foster what us Western Canadians call the Mathematical Processes (see p. 6) or what Americans might call the Standards for Mathematical Practice

(1) How many points is each position worth?
(2) What is the minimum number of first place votes you can get and still win the MVP?
(3) What is the best way to create two fair teams of five players?

If you take a minute with each of these problems, you will begin to appreciate their elegance. The first two appear to have a single solution, but many possible ways to approach its value. The last one is a matter of opinion, but questions of fairness prompt students to create arguments based on some type of numerate structure. While it is harder to see these arguments as explicit outcomes in a curriculum, they are undeniably mathematical. 

A closer look at each of the three questions provides insight into anticipated student milestones and possible lines of student reasoning.

(1) How many points is each position worth?

I think most students would typically start at the players who received the fewest votes to find some type of clarity in the sea of numbers, but the unanimous decision in the favour of Steph Curry makes the top of the table the easiest entry point. It is easily discernible that a first place vote is worth 10 points, but that doesn't move us any further into the field because he was the only one to get first place votes. It does, however, make the linear progression of 2-4-6-8-10 points a possibility. This is a reasonable conjecture, but quickly debunked by Kyle Lowry's point totals. 

Actually, If we assume that each denomination of vote carries an integral weight, Kyle Lowry's point total and distribution can result in only one possible value for a 5th and 4th place vote. We can then walk up the chart determining the value of a 3rd and 2nd place vote. 

Before this becomes too mechanical, take a look at the relationship between Kyle Lowry and Draymond Green. I had one student call this a "partial tripling". I may be tempted to eliminate the two intermediate lines in the table and ask students to determine the values of the votes with only these two lines. 

**Side Bar: What if the point values weren't whole numbers? Can you find a combination that works for Kyle Lowry? What about Kyle Lowry and James Harden? For how many players does your system work? **

(2) What is the minimum number of first place votes you can get and still win the MVP?

Students may use the answer from question one and determine the total number of possible points. They might make the assumption that a player would need over half these points to guarantee the win, but there are more than two people competing for the award. I imagine they would begin by securing every possible 2nd place vote, and seeing if it was enough. If it wasn't, they may also award 3rd, 4th, and 5th place votes. 

It is worth reminding students that a voter cannot vote for a player in more than one position, therefore the maximum number of votes any one player can receive (in any position) is 131. Also, each 1st place vote must be awarded to someone (and that someone cannot be you). Compounding the issue, you can only be voted for a total of 131 times. How can the points be distributed so you still win the MVP? How many players would even be eligible to receive a vote?

This question is a natural extension of the first; they make a fantastic pairing. 

(3) What is the best way to create two fair teams of five players?

This question has a distinctly different tone. Debates of fairness always seem to tie themselves in knots as students wade into the murky waters of data vs. experience. 

There have been four algorithms for choosing fair teams that I have heard students fight for: 

First is the straightforward one-for-one draft where the top two players become captains and take turns choosing the highest player available. This results in teams of 1-3-5-7-9 and 2-4-6-8-10. Opponents to this thinking are quick to point out that if you look at pairs of players all the way down the line-up, Team 1 has the advantage every time (1 v. 2; 3 v. 4; 5 v. 6 etc.).

Second, students aim to fix the problems in option one by imposing a snake draft. This means that Team 1 selects first, then Team 2 selects twice, then Team 1 selects twice...etc. This process continues until all players are taken. (Remember each team must have five players). This results in teams of 1-4-5-8-9 and 2-3-6-7-10. Some students point out that Team 1 still has the best player and Team 2 still must take the worst player. 

Third, students force Team 1 to take the worst player along with the best player. Each round, the team must choose the best and worst player available as a pair; this continues until all players are taken. Keep in mind, each team must contain 5 players. When it comes down to the 5th and 6th best players left, Team 1 takes the 5th and Team 2 takes the 6th. This creates a book end effect which results in teams of 1-3-5-8-10 and 2-4-6-7-9. In a slight alteration, some argue that a trade of 5th and 6th best players would make the teams more fair. 

Last, students argue that a combination of the snake draft and book end draft is best. Here, teams select the best and worst players available but in a snake fashion. Team 1 selects the best and worst, and then Team 2 selects the best and worst twice in a row etc. This results in teams of 1-4-6-7-10 and 2-3-5-8-9. 

Each of these arguments is supported by some type of numerate reasoning and communication based on their hypothetical ranks from one to ten. Some refer back to the raw data to justify certain approaches, but others argue that this years' results are not representative of the typical distribution of talent. Whatever the case, the context provides many natural (and curious) avenues into mathematical reasoning. When building, adapting, or choosing tasks for the classroom, the focus should be on the avenues for thought provoking activity and not the context in which you attempt to occasion it. 

A real-world context is the cherry on top. 


Sunday, May 8, 2016

(Min + Max) imize: A Classroom Game for Basic Facts

**this post was elaborated on in the May 2016 issue of The Variable from the SMTS.

This is a game that was adapted from a colleague in my department. He can't quite remember where it came from, but knows there was some influence from his undergraduate days. Nonetheless, he reinvented it to play with his Grade 9s, and this post represents yet another reinvention.
The game has a simple mechanism (dice rolling), and endless extensions to elaborate on and play with. These are both keys to a great classroom game (for me anyway). 
(Min + Max) imize practices basic operations within the framework of larger, conceptual decision making. While I rarely bring up the probability of the dice rolling, it is obvious that students are making decisions based on the chances of certain rolls being obtained. The idea is to practice basic skills and order of operations in a way that allows students to be active, numerate decision makers. 

The game:
Each game begins with a structure. The structure is composed of a series of blanks (for the 10-sided dice rolls) and operations linking the blanks together in various ways. The students can clearly see how many rolls of the die will occur (one per blank) and copy the structure before the game begins. When all students are ready, the die is rolled. After each roll, the value must be placed in a blank. It is illegal to wait until all the rolls are complete to make choices; it is also illegal to switch a choice after it is made. (Although, when students try this, it allows me to see their thinking very clearly). 

The goal:
Before each round, decide what the target should be. I usually play three rounds. In the first round, we try to maximize the result. In the second round, we try and minimize the result. In the third round, we try and get a result as close to zero as possible. Each round, the structure of the operations is altered and the die is re-rolled. After all the numbers are rolled, I circulate and collect "high scores" from students. 

The boards:
Be creative with your game structures. I have had the most success with 4-5 blanks. After that, calculations can become a hindrance. As your classes get better, division and exponents are a great way to stretch their thinking. I have provided a few of the structures I use, but most are improvised in the moments of teaching in response to student's needs.