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## Legendre Transforms on the PDM: Instructor's Guide

### Main Ideas

### Students' Task

*Estimated Time: *

### Prerequisite Knowledge

- Familiarity with the Partial Derivative Machine
- Familiarity with partial derivatives and their interpretations
- Familiarity with total differentials
- Familiarity with chain rule diagrams

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student
- A Partial Derivative Machine per group

### Activity: Introduction

### Activity: Student Conversations

- Make sure students understand the relation between the derivative(s) they were asked to find and the derivative(s) they found and also make sure students are also capable of reproducing the derivation of the relation of these derivatives.

### Activity: Wrap-up

Once groups have had enough time to find the requested partial derivative(s), convene the class. Ask (to the class) a group to describe how they went about finding the requested partial derivative. Once a group reports their process, ask (to the class) the other groups if they used the same process. If there are other processes, ask for groups to justify their process. If this does not result in a unanimous agreement upon a correct process(es), describe each correct process to the class and expose the errors in any incorrect process that was used. Repeat this for every requested partial derivative.

### Extensions

This activity is the fourth activity of the Partial Derivative Machine (PDM) Sequence on measuring partial derivatives and potential energy. This sequence uses the Partial Derivative Machine (PDM).

- Preceding activities:
- Quantifying Change: This small group activity introduces students to the PDM by asking them to determine how many measurable quantities exist within the system and how many of these quantities are simultaneously controllable.
- Isowidth and Isoforce Stretchability: In this small group activity, students are challenged to measure a given partial derivative with the PDM.
- Easy and Hard Derivatives: This small group activity asks students to write each partial derivative that can be formed from $x_1$, $x_2$, $F_1$, and $F_2$ and then categorize each as “hard” or “easy” to measure on the PDM.

- Follow-up activity:
- Potential Energy of an Elastic System: In this integrated laboratory activity, students use the PDM to determine the change in potential energy between two states of a nonlinear system.