Aunt Janice was extremely touchy about her age. When an impudent nephew was brave enough to ask her, she cunningly replied that she was 35 years old, not counting Saturdays or Sundays. So how old was she? -Another puzzle from Mo Page

Seven students came up with the following: There are 104 weekend days in a year, so she’s omitting (35 years)(104 weekend days/year) = 3640 omitted days, or nearly 10 omitted years. So Aunt Janice’s true age is 45.

Everyone’s satisfied. No other answers, no tugs of insecurity about the reasoning.

The puzzle: what do you say to them to get them to see that their answer doesn’t make sense? Remember, the goal is not to explain the correct answer well, the goal is to point out something that leads them to find the flaw in this one (or at least recognize that it’s flawed).

Hmm, if we now think she’s really 45, shouldn’t we count how many days she omitted based on that?

(Might lead to an infinite series approach to the answer…) ;^)

How about asking them to generalize their answer? What if she had said “50 years”? “100 years”? “2 years”? “1 year”? And, I think, as we get down to the smaller years, the model will wear and tear in a more obvious way. And if that doesn’t work, the nuclear option might be “What does the formula say if she says that she’s lived for 5 days?”

There’s always the apparently-unrelated problem approach, too, such as the classic “a man weighs half his weight plus 100 pounds. How much does he weigh?”. When they argue about 150 vs 200 pounds here (I hope!), after resolving that you can ask what this might have to do with the original question about her age, and perhaps they’ll see that their answer there corresponds to the 150 pounds answer here.

Dan

Strictly from the point of view of someone who has been deeply influenced by Polya…

How do we instill the habit of checking our guesses somehow? I hate the idea that my students might be totally reliant on my opinion as to whether an answer is correct or not. I have an inherent faith in modeling behavior – no matter how many times it does not work as wonderfully as I want it to – so I continue to suggest checking or to suggest trying another approach. However, I find that my students don’t tend to be self-starters in this regard.

Any wisdom out there on this one?

Usually my only hope is if I can get at least two different answers out there somewhere so the kids can argue with each other about which answer is correct and why. With a problem like this, I hope that if I give enough wait time during which nobody blabs an answer, I’d get at least some different ones out there. Even if they’re all wrong, in the arguing about which one is right I can get them headed down the right path, or at least I have a chance. When they all agree on the same wrong answer, it’s pretty tough! So if anyone has ideas about that, I’d love to hear them too.

Another possibility is to have your students figure out their age not counting Saturdays and Sundays. Once they do, take one of the fake ages and try to work it back to their actual age using their method. You can bill the last part as just summarizing the approach they used, and then you can act as surprised as everyone else when it doesn’t come out well.

Ooo! All the suggestions have been excellent but this one grabs me. In my classes (maybe this is universal?) anything that gets the students figuring out anything about themselves gets a lot more attention than anything that doesn’t.

Reminds me of lots of standardized test questions that revolve around how percent change works. Typical would be something like, “An item is marked up 25% from its cost and placed on sale. After it fails to sell, the store manager decides to place it on sale. What by what percentage should he reduce the current price to return to the original cost?” (apologies for trying to word that from memory)

The fundamental error, of course, is the failure to see that once you add or subtract a percentage of a base number to itself (e.g., +25% of x), the same percentage no longer represents the amount it did when you had the original base.

So how do you get kids to see this? There are many fine suggestions, and I think what you do here depends on what you want students to get from interrogating their own thinking. It feels so natural to me to see this in terms of fractions (easiest for me to do in my head) and say that knocking off 2 days per week is taking 5/7 of x (her age), so to get that age, I need to take the reciprocal, 7/5, of the reduced age. As 7/5 of 35 is 49, that should be correct. No muss, no fuss, no working with how many days in a month, year, decade, or century. And so easy to check: 49 *5/7 = 35.

What question might we ask to get students to even consider that 45 isn’t right, other than just checking the answer (nothing wrong with that, of course)? How about, “How many years has Aunt Janice been alive? Did you use that number in your calculations?”

Since they haven’t, not even as an unknown, how could they possibly arrive at the right answer by their method? They’d need to know the answer in advance to know how many weekends to deduct.

There’s no one right way, of course. Getting them to try their flawed approach on their own age is certainly engaging. But I hope that you tie this to a host of percent change problems until some real light dawns, because this sort of error is so ingrained in the heads of so many students (and adults) as to be deeply worrisome.

Update: they all acknowledged that their model doesn’t give the right answer. A couple people’s attention was caught by the fact that when we figured out our ages not counting weekends, the number of weekend days we subtracted from our true ages was different than the number we added back. But heads are still being scratched.

I assumed, like Michael (Pershan), that their model would break down as you try this with smaller and smaller values, but in a way, it doesn’t: their model is equivalent to truth = 9/7 * lie, and mine is truth = 7/5 * lie, and those two models behave pretty similarly.

Was there any further followup? Heads scratched = a natural question is why didn’t it work? On the edge of my seat!

Someone came up with a rather long argument that got them to 49. It was correct but convoluted, and I couldn’t persuade anyone to question it. So I think we’re left with people feeling as though a correct answer has been found so even if we don’t totally get it, we’re done. Once again I was on the horns of respect-their-process vs. model-clearer-thinking. We left that one there.

Interestingly I tried the same problem last night with my 10- to 12-year-olds at Math Circle and got into exactly the same bind, with exactly the same result. The difference was, when I asked them to figure out what a 14-year-old would say if he didn’t count weekends, they had a hard time with that, too — so no counterexample to work with! I thought it would be a gimme for them (they tend to have better intuition than my high schoolers) and so I left it very little time … I had to leave them with it, too, and it’s the end of the term!

So I’m 0-and-2 on this one, hoping the right moment will arise to bring it up again…