Review strategies for interpreting, estimating, and solving differential equations
Quick Lesson Plan
Students begin by working on the boiled potato problem FRQ (2017 AB4). This is then scored using the AP scoring guidelines. In the subsequent card sort activity, created by AP Calculus teacher extraordinaire Nancy Stephenson, each slope field card (there are 10) is matched to its unique differential equation card and a verbal summary of the solution curve or an important aspect of the solution.
Before assigning the boiled potato problem, read the solution carefully and try to anticipate where your students will have difficulties. This is a particularly challenging question and may work better as a group assignment for AB students. Go over the scoring guidelines and have students see how many points they would have earned. We strongly suggest not assigning a formal grade for this FRQ; instead, we want students to become familiar with the rigor of the AP test and learn some strategies for solving problems that may seem unfamiliar. One way to facilitate the FRQ is to allow for 5 minutes of individual think time, 5 minutes with a partner, and 5 minutes combined with another pair (think, pair, square).
To extend the card sort activity, use consumable materials and have students sketch a family of solution curves onto each slope field. Three of the differential equations (DE1, DE3, DE10) cannot be solved by separation of variables, but the remaining seven are good candidates for student practice. Create initial conditions for the seven solvable DEs to provide practice in solving for the constant.
As with every activity that encourages peer-to-peer conversation, listen carefully for common mistakes and errors of understanding and use those comments for whole-class discussion. Some students may still try to incorrectly separate variables by subtraction (for example, dy/dx = x + y) and others will struggle with the properties of logarithms and exponents: they can never practice their algebra skills too much!