I’m co-advising MathClub with another teacher. And two weeks ago, I presented the students with the idea of continued fractions and continued exponents. After a brief introduction, I gave them a really tough problem. Honestly, I didn’t expect them to solve it the same day, but they did.
One of the problems I used to motivate the more difficult problem we worked on was:
Find x if 2=x^x^x^x^x^x^… (ad infinitum). [problem 1]
Not getting into details, it turns out that x is the square root of 2.
BUT one student, a few days later, noticed that if you say:
find x if 4=x^x^x^x^x^x^… (ad infinitum) [problem 2]
and do the exact same analysis as you do for problem 1, you find that x also equals the square root of 2. Anyone spot the contradiction? (You basically are saying 2=4.) It turns out that problem 1 has a solution, but problem 2 doesn’t.
So it turns out that if you say k=x^x^x^x^x^x^x^… (ad infinitum), there are only certain values of k which have a solution for x. And this student decided this was something he wanted to work on.
So I let him loose… I thought the right way to think about this problem is though sequences, but this student decided to work graphically. And his method is ingenious, and he got the answer.
So next MathClub he’s presenting it.
And that’s amazing. Because he came up with the problem, and he solved it!