A collection of gathered thoughts and activities from MMSI (Massachusetts Math and Science Initiative) APSI (AP Summer Institute) AP Calculus AB Experienced workshop.
First activity–a variation on graph matching. Using name tags, students match derivative graphs with function graphs. I liked the physicality of the activity–I generally enjoy matching activities, and particularly liked the moving around twist.
Common student mistakes – some good discussions on individual error types, followed up by adding some notes that I can give to students. I picked up some additional thoughts on simplifying problems in free responses for students, which is nice.
Calculus Reasoning from Graphs – this was similar (I think) to something Mike presented last October, but I got more out of the activities this time for use in my classroom. We started out by looking at 2013 AB 4, which shows a polynomial looking graph over a small interval, and provides the areas between the curve and the x-axis. We had a nice discussion about an alternate form of the FTC as a sum instead of a difference, which ties nicely to point-slope in function form, and Taylor polynomials. This was also the only good (judged by my) argument I’ve ever heard for moving point-slope into function form–to make it easier to move into Taylor polynomials with even a little more comfort is rather lovely. I hope to use this packet, if not in class for a good several day lesson, for a Saturday session working with other students and my own.
Thought – Have students derive definitions for concavity using secant lines and tangent lines. Will secant lines always be above a concave up interval? Always be below a concave down interval?
The last thing we did was explore calculator use in calculus, and played with some places that calculators have issues graphing or solving problems due to the algorithms the calculator uses to perform calculus. Some cool examples to share with students: graphing y = sin(3x), then y = sin(47x) (a problem with calculator resolution); solving 0 = (x-2)^2 by graphing and finding zeroes (a problem with the way a calculator finds zeroes (looks for where the function changes sign)); evaluating the derivative at 0 and 1 of 1/(x-1)^2 (symmetric difference quotient issue); and the definite integral of (3+cosx)/x^4 dx from 1 to 3, and from 1 to 10^5 (a problem with the summation that the calculator uses to approximate based on intervals from 1 to 10^5). Cool stuff. Especially since learning that my students really struggled more on the calculator sections last year, and I need to help them improve this year–it’s always fun to have places that the calculator doesn’t work. I want my students to be thinking calculator users, not using a calculator to think.