I've always been interested in those people at amusement parks and fairs who guess people's weight or age or birthday month or whatever. One interesting question from that situation is which variable you should have the person guess for the best chance of winning. On one hand, birthday month feels nearly impossible for someone to guess by just looking at you, but the guesser does have a 1/12 chance of being correct. I can't remember the usual ranges for age and weight that let the guesser win, but it would also be interesting to think about how the amusement park sets those and if they're fair. To complicate things even more, how do social factors change what the guesser guesses (e.g. does the guesser under-guess age and weight for older people and women respectively, because that's what our society says is better?)
This problem is super-interesting:
http://nrich.maths.org/6957
I like that there is a lot of open-endedness to the solution and "correct answer." Unfortunately I am not sure what unit it might fall in because it involves a ton of different possibilities. Just a fun math problem? That's okay with me too!
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Update: I tried this task with a group of approximately 80 secondary math teachers (6th-12th grade). My version was slightly modified to (1) give it a little bit more of a hook and make it look pretty; and (2) obscure the task name so no one could google it... Teachers are sneakier than students. They shouldn't get to do what I did and just go straight to the sample student solutions!
I haven't looked at feedback from the session yet, but I personally enjoyed listening to what people came up with. There as a great deal of disagreement in the room about who should "win" and lots of different takes on a scoring system. Unfortunately we didn't have as much time as I wanted to, so I didn't get as much of an opportunity as I would have liked to push on some of the justification aspects, especially about why a scoring system is fair or ideal.
A lot came out about mean and standard deviation that I also didn't get a chance to make sense of. I wanted to ask people mean and standard deviation of what? What's their sample? Is mean or median a better measure of center? Part of that last question might rely on an assumption about guessing whole number weights. What happens when this task changes from discrete to continuous mathematics? I doubt the answer changes, but the questions you ask will definitely change.
