Order of operations, and odd/even games

[Joshua Zucker]

Last week's Numberplay blog (moderated by me) at
http://wordplay.blogs.nytimes.com/2012/02/06/numberplay-odd-or-eve/
has a problem that I've done with kids and with teachers' circles.

There are some fun strategy games that teach about odd and even, and
as you allow kids to mix multiplication and addition, maybe with a set
of parentheses, in the variations of the games that are discussed in
the comments, they need to practice their order of operations as well
as understanding that odd*odd = odd and odd+odd = even and so on.

What topics are happening in your curriculum these days, that make you
want an interesting problem that leads your students to think more
deeply about what they're doing?

Enjoy,
--Joshua Zucker

[Sean Corey]

I presented my students with a game which I got the idea from the book Understanding Game Theory.  This was in the "Coalitions and Distributions" section of the book.  

For groups of three kids, I assigned students cities A, B and C.  Each city needs to build infrastructure.  I posed the game in this way:  If city A builds alone, it will cost 60mil.  The Same for B and C.  If A and B form a coalition, then they can pay 40mil in total and both of their infrastructure gets built.  If A and C form a coalition, 50mil combined, and if B and C 50mil. Finally if all three form a coalition it would cost them 80mil combined and each city gets their infrastructure.  

I created the numbers in this game so that the AC coalition is the beast deal for C at minimum of 30mil, with room to go up 39 million and still beet out the triple coalition at 80 mil. This leaves B high and dry, so no B can barder with C.  Thus an endless system of bardering ensues as it did in my 7th grade classes, Mwahahaha! (evil math teacher laugh)

But, if you have students come up with their own numbers, then really good questions can arrise like "how can we choose numbers so that there is the most 'happy' equilibrium, the 'least' happy equilibrium.  Then they can answer "What would be a fair distribution?."  What would the utilitarian solution be? What would the most capitalist solution be?  

When I did this activity, the engagement was through the roof!  Unfortunately I did not do the greatest job of connecting clear mathematical content. This problem houses:

inequalities

substitution?

...

Any advise on this activity?

Regards,

Sean Corey