Had a fun time yesterday showing up at the end of the NCTM conference in Boston. What was originally a plan to grab lunch with Fawn Nguyen and Dan Anderson turned into a full day of meeting tons of people I’d only known online.
Over drinks at one of the bars at the conference hotel, Chris Hunter and Fawn showed me a really cool problem from Chris’s blog:
The problem is pretty easy to state – you and two friends go to the store to buy shoes. You have a “buy 2 get one free” coupon that allows you to get the lowest price pair out of 3 pairs for free. The question is what is the fair way for the three of you to split the savings?
I was so excited to try out this problem with the boys today that I stole the napkin that Fawn was writing on!
It seemed like the best way to go through this problem was with each kid individually. I started with my younger son. He had a little bit of trouble understanding the problem (so this video goes about 7 min) – the cost savings combined with the free item confused him, for example. However, with a few little clarifications he was able to get to an answer that he thought was fair.
Next up we looked at a similar problem with different numbers. These numbers present a new issue to deal with if you want to split the total price equally. This second problem also served as a great way for my younger son to get a little more clarity on some of the previous parts of the problem that had confused him.
Next up was my older son. His initial focus was on everyone paying the same amount, but after thinking about it for a little bit longer the equal split idea started to bother him. He wasn’t sure what a “fair split” meant. He thought for a while about other fair ways to split the price and eventually found the idea of splitting the savings. That thought process shows what I really like about this problem – lots of opportunities for thinking here and no obvious “right” answer. At the end, though, he thought splitting the total price equally was the most fair.
The neat thing about my son’s conclusion in the first problem is that it set up the next problem perfectly. If we split the total price equally in the second problem, there’s a strange issue for one of the three people. It takes him a while to notice the problem, but when he does notice it he thinks that the “splitting the savings” here is the fair way.
So, definitely a great problem for getting kids to talk about math. I really like the idea that different people are going to have different ideas about what is fair here, and I imagine those different ideas would lead to some really fun debates in a classroom setting.
Mike's Math Page 
Aww, I was waiting on the distribution of a percentage savings but it never materialized (ie. pay 75% and 80% of original value respectively). Probably not intuitive at this stage. Would have been great to have done this last week and discuss the fairness of a flat tax. :o)