Lesson Overview: As you’ve already read, this is a self-contained module for calculus students. One-dimensional kinematics is typically covered in beginning calculus courses, whether the level is high-school honors, AP, or collegiate. This module could take 1-2 days depending on the readiness of the students and their allotted time. I’ve included a guided worksheet for the students to complete as they go through the module. See the next page for copies of this, as well as the key.
The module's primary emphasis is on the notions of rate of change, critical points, and a graphical understanding of the differences between position, velocity, and acceleration. Moreover, because this module was completed as a result of a research experience, I’ve dedicated a portion of the site explaining how applied math connects with real-world phenomena, as well as other sciences. While this content isn’t necessary for an understanding of the essential ideas, I believe that it’s important because students should be exposed to the nature of research in math and science! For more information regarding the nature of mathematics, I encourage you to check out the official benchmarks.
Essential Calculus Ideas: Show/Hide
- Motion in one dimension (along any line) may be modeled on the Cartesian plane as time versus displacement from an origin.
- Given a position function for an object, the slope of the function at a point (found by taking the derivative) gives the rate at which the particle is moving, or its velocity.
- Speed is the absolute value of velocity.
- Given a function for the velocity of an object, the slope of the function at a point (found by taking the derivative) gives the rate at which the velocity is changing, or its acceleration.
- The critical points of a function for position or velocity indicate possible changes in the values (from positive to negative, or vice versa) of velocity or acceleration, respectively.
- One may “read” the graph of a function for position or velocity to give information about velocity or acceleration.
NCDPI/AP Objectives: Show/Hide
- 2.03, Interpret the derivative as a function: translate between verbal and algebraic descriptions of equations involving derivatives.
- 2.05, Interpret the second derivative: identify the corresponding characteristics of the graphs of ƒ, ƒ', and ƒ".
- 2.06, Apply the derivative in graphing and modeling contexts: interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
Offline Lessons: If you don't have access to the necessary technology, I've also included complete lesson-plans that form a unit on 1-D kinematics. Both lessons are designed for a 90-minute block schedule, and each includes a student note-sheet.