Teaching Tips

We investigate slope fields and differential equations today and our students love, love, love these problems! Our instructional approach for solving DiffEq’s was straightforward: students followed the SISIS method (see Topic 7.7). Constructing a slope field and solving a differential equation are reserved for FRQs and allow for the opportunity to connect multiple representations of a single function within one problem. These skills are almost always presented together as students work to find the one correct particular solution from the numerous possibilities shown on their slope field. 

 

Because we were so close to the AP test day, we simulated the exam experience by having students work through the entire problem in 15 minutes. Confidence was high for this FRQ. Not only did students feel well-prepared for the calculus, they were familiar with the scoring rubrics for DiffEq’s and knew exactly how to earn points as they worked!

 

When constructing a slope field on the AP Test, prepare students to:

• Remember that this is tedious work: only a few slopes will be required and students must place their marks ON the indicated coordinates and nowhere else

• Draw their slope lines in relation to each other: a mark representing a slope of 3 must be steeper than a mark representing a slope of 0.5

• Refrain from drawing vertical slope marks: these are not a part of a slope field

• Look for trends: if one mark is wildly different than its neighbors, confirm your results

• If told to include point (a, b) in their solution curve, be sure to draw a curve through that point!

• ENCOURAGE careful work; attention to positives/negatives; using only the given coordinate pairs

• DISCOURAGE vertical slope marks; drawing possible solution curves outside the given grid; attaching arrows to the ends of their solution curve

 

When solving a differential equation on the AP Test, prepare students to:

• Use the SISIS method for solving!

• SEPARATE variables to align with the differentials (dx, dy, dt, etc.) and earn 1 point

• INTEGRATE both sides for 1 point; include +C for another point

• SOLVE for C and earn 1 point just for utilizing the initial condition

• ISOLATE the dependent variable and remember to show all possible solutions

• SELECT the solution satisfying the initial condition AND the original DiffEq to earn 1 point

• ENCOURAGE including +C even if they are not 100% confident of their antiderivatives; attempting to use the initial condition in their antiderivative expressions; using absolute value when solving a natural log expression; considering  domain restrictions when choosing their solution; comparing their solution to the slope field, if time permits

• DISCOURAGE skipping any steps in the SISIS method because students might “just know” the antiderivatives