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## Quantifying Change: Instructor's Guide

### Main Ideas

Estimated Time: 15 minutes

When first introduced to the Partial Derivative Machines, the central system was hidden from students through the use of a “black box”. With only the knowledge that there were two strings extending from this box, students were asked to determine:

• The properties of the system that can be controlled.
• The properties of the system that can be measured.
• The number of independent properties of the system.

• ### Prerequisite Knowledge

• Familiarity with Derivatives
• Familiarity with Integrals

### Activity: Wrap-up

Students worked briefly in groups to answer this prompt and then were brought back together for a class discussion. During the wrap-up discussion, students listed a number of controllable properties including the position of the central system relative to the center of the board, the forces applied to the system, and the amount the system was stretched in either direction. Students decided it was possible to measure $x_1$ and $x_2$ by taking values for the positions of the flags, and to measure $F_1$ and $F_2$ by noting the mass hung from the relevant string.

Many students did not realize however that the tension in a particular string is not equivalent to the weight hung from that string if the corresponding knob is locked since the mass becomes irrelevant when the string is pinned down. Most students also determined that only two of these properties could be controlled independently and that manipulating a pair of parameters caused a responsive change in the other parameters.

### Extensions

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

• Follow-up activities:
• 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.
• Legendre Transforms on the PDM: In this small group activity, students get a chance to work with physical analogues of Legendre transforms.
• 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.

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