Review Topics 8.1-8.3
Quick Lesson Plan
Two outstanding FRQs will be used as our review of Topics 8.1 - 8.3. Students will work through 2019 AB1 (which serves as a nice review of 8.1 average value and 8.3 rate in/rate out concepts) and 2016 AB2 (for a review of 8.2 particle motion strategies).
After reviewing the general directions for free response questions (for example: pen vs. pencil, calculator use, using calculus notation and vocabulary, confirming that the original question has been answered), direct students to work independently for ten minutes on 2016 AB2. Then, allow students to work with a partner for an additional 5 minutes to correct their notation and numeric values or to improve justifications. Distribute a copy of the Scoring Guidelines to each student and have them self-score their work.
A whole class conversation may reveal common errors or confusion which can be addressed immediately.
Now that students have warmed up, distribute the second FRQ (2019 AB1) to each student for 15 minutes of independent work. You may choose to distribute samples of student work (provided by College Board) along with the Scoring Guidelines and have students practice scoring the samples before scoring their own responses. Another option is to collect student work and grade the FRQs using the Scoring Guidelines and the Chief Reader’s report for this question. Misconceptions and errors can then be addressed during tomorrow’s review.
2016 AB2: In general, students should reference the given function(s) when explaining or justifying their responses. So, in part (a), discussing v’(4) is more appropriate than a(4), especially without writing a(t) = v’(t).
Encourage students to use the term v(t) as the integrand in part (c) to save time and avoid copy errors.
2019 AB1: Part (a) will reward the careful reader (round to the nearest whole number!) and the student who uses E(t) for the integrand expression: they will save time and avoid copy errors by simply writing E(t). Forgetting to divide by 5 was the most common error when finding average value. Students intently calculate the value of the integral, but then forget to find the average value over [0, 5]. Using the candidate test in part (c) is rich in notation, but still requires students to determine when E(t) = L(t).
2016 AB2: To determine if a particle is speeding up or slowing down (part a), students may forget to compare velocity and acceleration. In part (b), the justification must include a sign change for v(t). Total distance is computed with the absolute value of only the integrand --- not the absolute value of the entire integral expression!
Part (c) will pose a challenge for many students. Remembering that E(t) and L(t) are already rates removes the need for finding E’(t) and L’(t). The desired rate expression is actually E(t) - L(t) and students must determine when that quantity changes sign from positive to negative. Some students will note that we are working with a closed interval in part (c) and use the candidate test. “Is the rate of change… increasing or decreasing…” may seem confusing, but actually commands students to find the rate of change of E(t) and L(t): E(t) and L(t) are already rates of change and E’(t) and L’(t) will reveal to students the increasing or decreasing nature of those functions. Caution is needed here: the rate of change of a rate of change is designated by only the FIRST derivative!