The problem above came across my twitter feed this morning courtesy of John Joy (@johnjoy1966) along with the suggestion that this was a problem from a trig unit. John also questioned who this problem would be appropriate for. I told him I would feed it to my wizards in AP Calculus BC. I also had a class coming in right after John posted it so I did not see any of the conversation – this way I could present it to my class with no prejudice about what to say. When they came in the next period I had the problem from the tweet up on the screen with no other support. I simply said that this seemed like an interesting problem and I had not had time to try it myself. I handed out the whiteboards to each desk group – this was their suggestion! – and I got out of the way. I heard them talk about the function being odd so that they knew f (-3) right away. One group found f (6) by imagining it as f (3 + 3). This meant, of course that they also knew f (-6). Progress, right? But nothing about knowing the values of f (1), f (2), or f(3). I asked if they wanted to hear a hint and three students quickly waived off that notion. After another couple of minutes I went to the board and started writing what we seemed to know about the function. I wrote f (3) = f(1 + 2) and wrote out what the definition of the function suggested. We got an ugly expression for comparing f(1), f(2), and f(3) to each other but it was not promising. I was itching to give them a hint but they were holding me off. One of my students – thinking out loud – wondered if this might be a periodic function based on the values we knew on the board. Another group suggested that it might be a sine function. I hopped at this – another example of how I need to work on developing a poker face of some sort. The group backed up a bit and suggested that they were kind of joking, but I buoyed them up by reminding them of the periodicity suggestion. I finally gave them a vague clue – one too vague to have helped them at all. One of my students during a class warm up a few days before had the back of his book open to a series of formulas and review facts from their study of trig. I reminded the class that I complemented him on that and I pointed out the similarity of the given function to a trig identity involving the tangent function. The kids were a bit flustered claiming that no one remembers these formulas but they sealed the deal right away once they had this fact in hand.
So, what did I learn from them today?
- I need to work on my poker face.
- I need to stop giving clues, they are too good to need them.
- This group of students is super persistent and creative.
- The small desk groupings AND the randomization every Monday seems to be working.
- The whiteboards give them space to ‘think out loud’ and effectively share ideas.
Man, a terrific day in Calculus thanks to my wizards and my virtual friends who prod my brain with their great problems.