Particle motion makes up about 14% of the AP Calculus Exam. There are generally five or six multiple choice questions and one free response question dedicated to this important topic. In the AP Curriculum, students see particle motion in two different units: first from the perspective of derivatives in Unit 4, and then from the perspective of integrals in Unit 6. In this article we’ll focus on particle motion from a derivative perspective (Unit 4).
We’ve chunked the content into three main categories: what students need to know, what students need to be able to do, and what students need to understand.
With particle motion, like most other big Calculus topics, we have to take the long view. We can’t expect students to have all of these skills and understandings after just a day or two, otherwise they will be forced to memorize lists of interpretations and justifications without the conceptual understandings that make that content memorable and meaningful. Many opportunities for discussion and sense making, meaningful problems, and spaced practice with increasing levels of complexity are the best methods for making sure students have a firm grasp on particle motion.
Resources for teaching particle motion
We introduce particle motion with “The Lovely Ladybug” EFFL lesson. In this activity, students have a quadratic position function and draw the graph of the linear velocity function. They explore positive, negative, and zero values of the velocity and debate how to determine whether the ladybug is speeding up or slowing down. In the final question, students are given the velocity function of a worm and are asked to consider what the worm’s position function could be. Without knowing anything about integrals or antiderivatives yet, students are able to “work backwards” and use some trial and error to find a possible position function. This plants the seed for Unit 6 when students learn about antiderivatives and determine an object’s displacement and total distance using integrals. Check out the lesson as well as a complete answer key and teacher notes below.
On the next day, we use Bryan Passwater’s Big 10 activity to give students more purposeful practice applying particle motion principles in various representations.
There are multiple released free response questions that you can have students work on in class or assign for homework. Here are a couple of our favorites. Note that students won't yet be able to do all parts of these questions.
2011 AB 1 (analytical: parts a and d)
2008 AB 4 (graphical: parts c and d)