Calculus students displayed their work on a challenging optimization problem. They were asked to plan a design to hang “something” from “somewhere”, and then determine the least amount of rope or wire needed.
Some of the things they chose to hang included a Williston medallion from the Golden Gate Bridge, a disco ball from Eiffel Tower, a circus hoop from the circus tent, a tire swing from a tree, and a target in a hockey goal. They needed to plan how far apart the hangers would be and how far down their item would hang. They then used Calculus to find the minimum amount of rope or wire needed for the project. Their presentations needed to include all of their calculations and drawing to support their work and solutions. They also needed to build a model to represent their solution.
A few days after each AP Calculus BC exam, the College Board releases the free response questions from the exam. They don’t release their very succinct answer keys for a few more weeks… so… I had my students make their own answer keys as well as screen recordings of their solutions!
All 2017 released free response questions and answer keys are online right here. Questions and answers for past years can be found right here.
Teaching Calculus to seniors and a few juniors, I feel an obligation to help move them toward independence and self-sufficiency in their learning. I want them to learn to support themselves as learners and know how to reach out for assistance. I use two primary methods to this end.
First, I provide full solutions for all homework. Students are expected to use these solutions to check their work as they complete each problem to be sure they not only have the correct answer, but more importantly, that they have supported their work appropriately. The other way they can use these solutions is as a hint on how to start a problem if they just need a little help. No one should come to class with a blank homework saying they did not know how to do the work. With this, they know when they need help and are expected to ask for it.
My second strategy for student increased independence is to have all students at the boards at the same time to do problems together. By being visible, at the boards, they can and should look to others around them to confirm they are headed in the right direction. Each student becomes a source of information for everyone else. Students who might not take the lead sitting at their desks are now asked for help by their peers. Again, no one is left unable to start a problem. Help is all around.
For the last few days in Calculus, with help from Mr. Roe in the Art Department, my classes built models as part of a project exploring solids of revolution. Partners were assigned an equation, asked to sketch and graph that equation and then find the volume of the solid that would be created by revolving that graph around the x-axis. They needed to sketch and estimate the volume if filled with only four cylinders, then find the estimated volume if filled with eight cylinders using a computer generated model. Using Calculus they were able to find the actual volume and compare it with their estimates. Next they needed to create a three dimensional model of their solid using wire and foam core. On the last day of classes they presented their project to their classmates. The projects were on display in the Reed Center in time for graduation for their families to see.