PDM Variables

1. This small group activity is designed to help students become familiar with the Partial Derivative Machine (PDM), a mechanical system which allows students to measure partial derivatives and determine various physical quantities, including potential energy.
2. Students use the PDM to determine which properties of the physical system they can control and which ones they can measure.
3. The whole class wrap up discussion focuses on the fact that not all of the properties of the physical system are independently controllable, and leads students to think about what is being held fixed when they measure a derivative.

Partial Derivative Machine

1. This small group activity is designed to provide students with a means of measuring derivatives using ratios of small changes.
2. Students measure partial derivatives on the “Partial Derivative Machine.”
3. The whole class discussion focuses on how to represent partial derivatives in multiple ways and how to experimentally measure derivatives.

Internal Energy of the “Partial Derivative Machine”

1. This small group activity is designed to provide students with the means to the potential energy of an elastic system in one dimension.
2. Students use the “Partial Derivative Machine” to experimentally measure quantities in order to compute the potential energy of their system.
3. The whole class discussion focuses on how to experimentally measure integrals and how to represent integrals in multiple ways.

Isowidth and Isoforce Stretchability

1. This small group activity is designed to show students how to calculate derivatives using small differences while paying attention to what is held constant.
2. Students use the Partial Derivative Machine to measure partial derivatives while keeping different variables of the system constant.
3. The whole class discussion focuses on how to represent derivatives in multiple ways, experimentally measure derivatives, and show that the “thing held constant” both has physical and mathematical consequences.

Easy and Hard Derivatives

1. This small group activity is designed to help students become familiar with the Partial Derivative Machine and how to think about measuring derivatives.
2. Students practice thinking about which derivatives are easy and hard to measure in the context of the PDM.
3. The whole class discussion focuses on becoming familiar with the PDM, how to think about derivatives, and which derivatives are easy to measure and which are hard.

PDM Legendre Transformations

1. This small group activity is designed to help students learn about Legendre transformations and thermodynamic potentials.
2. Students work in small groups to investigate ways of describing a system as a function of different variables.
3. The whole class wrap-up discussion emphasizes the need for Legendre transformations and thermodynamic potentials on the PDM and their connections to their thermodynamic counterparts.

Legendre Transforms on the PDM

1. This small group activity is designed to gives students a chance to work with physical analogues of Legendre transforms.
2. Students use their PDMs to measure a list of partials that are not directly physically measurable.
3. The whole class discussion focuses on how to know when a Legendre transform is necessary.

Potential Energy of an Elastic System

1. This integrated laboratory activity is designed to ask upper-division undergraduate students to measure the change in potential energy in an elastic system (PDM) between two different states.
2. Students use the Partial Derivative Machine (PDM) to verify experimentally that the forces and dimensions of their system are state variables as well as measure the relationships between these quantities to compute the potential energy of their system.
3. The whole class discussion focuses on the meaning of integration of discrete experimental data.

Upside Down Derivatives

1. This small group activity is designed to provide students with a means of experimentally verifying relationships between partial derivative expressions.
2. Students use the Partial Derivative Machine (PDM) to measure two “easy” derivatives that are mathematical reciprocals of each other in order to demonstrate a relationship between them.
3. The wrap up discussion focuses on helping students realize that when the variables in the numerator and denominator of a partial derivative are switched, and the same variable is held constant, that the numerical value of the derivative is simply the reciprocal of the original quantity.

Cyclic Chain Rule

1. This small group activity is designed to help students practice measuring derivatives with the Partial Derivative Machine (PDM) and help them make connections between mathematical expressions and physical systems and measurement.
2. Students are given a written representation of the Cyclic Chain Rule and are asked to assess whether the dimensions make sense and to use the PDM to verify the expression.
3. The whole class discussion focuses on the results of the measurements and leads to the instructor introducing the concept of the “Cyclic Chain Rule”.

Deriving Change of Variables

1. This small group activity is designed to provide students with practice in using a change of variables procedure for partial derivatives and total differentials.
2. Students use the variables from the Partial Derivative Machine (PDM) to practice the mathematical technique of change of variables when given a partial derivative that cannot be explicitly measured.
3. The whole class discussion focuses on how students manipulated the mathematical quantities to arrive at a total differential from which they could successfully extract the requested derivative.

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