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Functional dependence problems - General
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Highlights: A collection of 16 problems that focusing on functional dependence using geometry and exponentials, often in a biological context.
Abstract: One of the most important ideas IPLS students need to learn is that equations are not just calculational tools; they represent relationships between physical variables that change together (co-vary)—their functional dependence. Understanding this sort of relationship is rarely
emphasized in introductory mathematics classes, and students who have not yet learned to blend conceptual ideas with mathematical symbols may not see the relevance and power of this idea. This unit offers a collection of readings, activities, and problems that can help students learn to use functional dependence productively.
Resource Types: Homework, Restricted access
Commercially available in:  ExpertTA
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Functional dependence problems - General

Components of cells.pdf

Cylinder Volumes.pdf

Exponential parameters.pdf

Finding a small tumor.pdf

Finding the right dose.pdf

How big is a worm.pdf

Molecules in a cell.pdf

Mass-specific metabolic rate.pdf

Pie are sometimes square.pdf

Scaling cubes.pdf

Pizza sale.pdf

Using functional dependence.pdf

Thinking about functional dependence.pdf

The exponential function.pdf

Tumor growth.pdf

Components of cells.url

Cylinder volumes.url

Exponential parameters.url

Finding a small tumor.url

Finding the right dose.url

How big is a worm.url

Mass-specific metabolic rate.url

Molecules in a cell.url

Pie are sometimes square.url

Pizza sale.url

Scaling cubes.url

Scaling up.url

The exponential function.url

Tumor growth.url

Thinking about functional dependence.url

Using functional dependence.url

INSTRUCTOR GUIDE


IMPLEMENTATION

Equipment required:  Computers / software

Specific equipment needed:  Web access is needed for the homework problems for students.

Basic implementation tips & tricks:  These materials represent a thread rather than a unit. It contains materials (readings, lecture slides clicker questions, and group activities) that can provide a unit at the beginning of a class, but for this to work as intended, functional dependence needs to be woven into class activities throughout, both whenever new concepts are introduced, as part of the
standard analysis of new complex equations, and calling it out explicitly when it is used in a derivation or explanation in class. In addition they need to be assigned problem solving that uses functional dependence so that they are not only shown but have to do themselves.

How does this resource fit into the flow of your course?  See the "Basic implementation tips & tricks"

PEDAGOGY

Pedagogical approach:  Collaborative problem-solving; Conceptually-oriented activities; Context-rich problems; Mathematically-focused activities

Skills / Competencies:  Dimensional analysis; Functional dependence; Multiple representations; Intuition building

What insights or realizations do you hope students gain from this resource?  (1) That equations represent not just calculational tools for finding numbers, but express relationships among physical quantities. (2) That even if they don't know specific values, an equation can tell how a quantity will change when something else in the equation changes. (3) That different functional dependences (surface vs volume, for example) has powerful implications for biology.

Why is this resource useful to life sciences students?  IPLS students tend to see equations as calculational tools rather than as a way to express relationships among parameters and variables. This leads them to miss two important ideas: (1) that some changes are more important than others and functional dependence is the key to seeing that difference: (2) that functional dependence implies scaling — for example surface vs volume effects — an idea that has considerable power and implications in biology.

DISCUSSION

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SUBMISSION DETAILS


Copyright:   2023 Edward Redish

License:   CC BY-NC-SA - Attribution, No Commercial uses and Share Alike. Derivative works must have the same license.

Last Edit Date:  August 22, 2023

Vetted Library Publication Date:  July 1, 2022

Submission Date:  December 28, 2021

Version: 
Version 9, August 22, 2023
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