This is one of the most tragic math problems I've ever seen. (Click for larger.) Not because it's awful, though it is, but because the awfulness conceals something amazing. I mean, how great is it that we can drop a rock in a well and the sound of the splash tells us how deep the well is. That's wizardry!
I find it completely amazing we get to offer that power to our students. If my goal were to conceal that amazingness, though, to ensure my students would be less interested in mathematical wizardry thanks to my efforts, I'm not sure I could do any better than this problem.
- The student experiences act one and act two at the same time. Act one is supposed to hit you in the gut; act two in the head. The only reason your textbook tries to do both at the same time is because printing the same problem on two different pages is logistically impossible. Luckily, you aren't bound by the same constraints.
- The problem starts in the second act. And what a second act. Your students have no idea why they're wading through that long, thickety paragraph outlining the tools, information, and resources (act two) they'll need to solve the hook (act one) which shows up long after they've stopped caring.
- And what a hook. Seriously, could someone please explain to me which interest group or political constituency is served by slurring what should have been concise, obscuring what should have been clear, and jargoning what should have been conversational. Seriously, how would a human phrase that hook? Would a human need twenty-six words?
- The act one visual is cheap. Again, we're dealing with cheap clip art here only because of the constraints on an industry that's taking on water. Don't go down with that ship. Can you think of a better visual, one that would make students wonder, "Wow. How deep is that?" without you lifting a finger?
- The act three payoff is weak. Imagine all the intensity of the final assault on the Death Star in Star Wars. A planet's survival hinges on an unimaginably long shot. Luke takes that shot as the clock winds down, a shot right at the guts of the Death Star. What if at that moment we cut to some Rebellion functionary announcing in a slow monotone, "The Rebels were successful. They destroyed the Death Star." That's what it's like to read the answer to a visually compelling problem in the back of the book. Show that thing explode.
It turns out that Hollywood occasionally makes math problems for us. Click through and have a look.
With Brendan Fraser, you get a fun check on your own answer and an explanation of why his team even cares how deep the cave is. With the Descent team, you get a much deeper cave and a stronger separation between the first and second acts. Both represent massive improvements over our status quo.
To be clear, I'm not saying you can just play act one and two and your students will trot merrily to an answer in act three, deriving that thorny equation for projectile motion all on their own while stopping periodically to smell the constructivism flowers. I'm not saying that. This problem is tricky and will likely require lots of help on your part. What I'm saying for sure is that it makes no sense to offer that mountainous paragraph of helpful text without your students knowing (to say nothing of caring) why you're offering it.