A quick reflection here before I wake up my kiddos.
Yesterday in BC Calculus we had onto of our weekly problem days where we (mostly) put aside our current Calculus work and look at interesting problems that may or may not involve any Calculus at all. Here is problem #1 from yesterday (a problem I borrowed from @bretbenesh.
A mountain climber is about to climb a mountain. She starts at 8 am and reaches the summit at noon. She sleeps at the top of the mountain that night. The next morning, she leaves the summit at 8 am and descends using the same route she did the day before, reaching the bottom at noon. Why do you know that there is a time between 8 am and noon at which she was at exactly the same spot on the mountain on both days? We should not assume anything about her speed on either leg of the trip.
One of the things I enjoy most in teaching is seeing/hearing different ways of attacking a problem. When I read this problem I immediately sketched a height v time graph with the base of the mountain and 8 AM as the origin and the top of the mountain, noon as some arbitrary point in the first quadrant. A wiggly sketch connected the points. Day two has a y-intercept of top of mountain, 8 AM and an x-intercept of bottom of mountain, noon. No matter how I connect these points the sketch intersects my other sketch and I see the reason why. I’m surprised by this discovery, but I see it. In each of my 2 BC classes yesterday there was one student who saw through this problem and explained it away so quickly that I was wowed. In each case the student immediately switched to thinking of 2 people rather than one. If one person starts at the top at 8 am and walks down while the other starts at the bottom and walks up then they must pass each other at some point! Simple, clean, elegant. It’s fun to learn from your students, isn’t it?