So, in my BC classes we are wrapping up our tour of integration techniques. It’s pretty easy when you are convinced that integration by parts is the strategy to use, or when you know you are supposed to use a trig substitution, etc. Little parcels are easy enough to deal with. Throw them all in a bag at once and choose? Much much more challenging. Yesterday, in our 40 minute classes, each of my two sections of BC Calc made it through two problems and I could not be more proud of them. They fought, they tossed out ideas, they stuck through some thorny algebra. They critiqued each other’s ideas. They questioned mine. I tried – I really did – to give them space and let it unfold. Other than one idea that I knew would lead to pain, I did my best to let them run the conversation. It was the kind of day that justifies – at least in my mind – our decision to have BC as the second year calc class. In a one year track these kids would not have days like this where they could just play with ideas without regard for the clock. The problem that was the real winner is below (if my cut and paste graphic works right)

Edit – Image pasting is not my strength right now. Sigh. The challenge at hand was to integrate the fraction dx/(x^(2/3) + 3x^(1/3) + 2)

One student in each class suggested completing the square and that was pretty thrilling. The first one even pushed a step or two through on a trig substitution involving secant. That’s where I intervened because I was pretty sure that this path would lead to pain. We looked at GeoGebra and tried to work backwards from its answer after we went down the partial fractions path. Man, what a good day and I was fortunate enough that one of my colleagues came to visit yesterday morning. She was their AB teacher last year so it’s possible that they stepped up their game for her. If that’s true, I’ll have to enlist her for future challenge days.