1-030-EyeModel-ModelingScenario

05 Jul 2022 | Contributor(s): Brian Winkel | doi:10.25334/FEHY-8R04

We offer a mathematical modeling experience using differential equations to model the volume of an intraocular gas bubble used by ophthalmologists to aid the healing of a surgically repaired region of the retina.

1-015-Torricelli-ModelingScenario

05 Jul 2022 | Contributor(s): Brian Winkel | doi:10.25334/5822-B846

We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.

1-005-OilSlick-ModelingScenario

05 Jul 2022 | Contributor(s): Brian Winkel | doi:10.25334/HHF9-GM75

We describe a modeling activity with difference and differential equations which enlightens students on the model building process and parameter estimation for a linear, first-order, non-homogeneous, ordinary differential equation.

1-007-AntTunnelBuilding-ModelingScenario

05 Jul 2022 | Contributor(s): Brian Winkel | doi:10.25334/DQRP-G032

We pose the prospect of modeling just how long an ant takes to build a tunnel. With a bit of guidance students produce a model for the time it takes to build a tunnel of length x into the side of a damp sandy hill.

1-001-MM-DeathAndImmigration-ModelingScenario

05 Jul 2022 | Contributor(s): Brian Winkel | doi:10.25334/XAHX-QD79

We describe a classroom activity in which students use M&M candies to simulate death and immigration. Students build a mathematical model, usually a linear first order, difference or differential equation, collect data, estimate parameters, and compare t

3-065-UpDown-ModelingScenario

12 Jun 2022 | Contributor(s): Brian Winkel | doi:10.25334/KCN1-6M41

We model the height of a launched object which is subject to resistance proportional to velocity during its flight. We ask questions about the motion as well, e.g., highest point or apex and terminal velocity.

5-030-LinNonHomoSystemSol-TechniqueNarrative

28 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/SXN2-EX75

We offer strategies for solving linear systems of nonhomogeneous differential equations using a conjectured solution strategy for a system of constant coefficient, linear, n

5-012-LinearSystemConjecture-TechniqueNarrative

28 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/HPY8-MX61

Students go from the solution for y'(t) = k*y(t) to a natural extension to the solution conjecture of a system of two constant coefficient, homogeneous, linear differential equations introducing eigenvalues and eigenvectors through student discovery.

7-005-OverViewLaplaceTransforms-TechniqueNarrative

28 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/4YYG-GD42

This is a specialized overview of Laplace Transform application to solving differential equations in Mathematica.

8-002-TrigSumRepresentation-TechniqueNarrative

28 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/FKZS-6A65

Students discover how to represent functions as sums of trigonometric functions and the value of such representations in many fields. This is an introduction to the study of Fourier Series.

1-003-IntroNumericalMethods-TechniqueNarrative

27 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/H8GA-T005

We develop elementary approaches to numerically solving first order differential equations with Euler's Method, Improved Euler's Method and develop these geometrically to compute numeric solutions and compare them to analytic solutions.

1-002-IntegratingFactor-TechniqueNarrative

27 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/Z3AQ-TP11

We develop a strategy to solve first order differential equations by transforming one side of the equation to the derivative of a product of two functions, thereby making it easy to antidifferentiate that side.

1-001-SepartionOfVariables-TechniqueNarrative

27 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/P07G-1Z58

We discuss strategies to solve first order, ordinary differential equations with mathematical models when the variables may be separated.

5-001-LSD-ModelingScenario

27 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/857M-FJ65

We describe the use of a two compartment model of a linear system of first order linear differential equations to model lysergic acid diethylamide (LSD) in the body. We provide the data from the literature.

1-030-EyeModel-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/5J93-AH54

We offer a mathematical modeling experience using differential equations to model the volume of an intraocular gas bubble used by ophthalmologists to aid the healing of a surgically repaired region of the retina.

1-025-MixingItUp-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/1YTQ-7Z66

Students build three different models for levels of salt in a tank of water and at each stage the level of complexity increases with attention to nuances necessary for success.

1-022-SpreadOfTechnologies-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/Z18P-FZ63

We examine plots on the spread of technologies and ask students to estimate and extract data from the plots and then model several of these spread of technologies phenomena with a logistic differential equation model.

1-023-RumorSpread-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/TJPS-B862

We use a newspaper report on the spread of a rumor based on shares of articles on the Internet over a 5 day period to demonstrate the value of modeling with the logistic differential equation.

1-020-IceMelt-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/4GMG-0Y75

We offer up the claim of a store catalog that its ice ball mold allows users to ``. . . make ice balls that outlast cubes and won't water drinks down.'' We ask students to build a mathematical model to defend or contradict this claim.

1-017-DiseaseSpread-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/8BJF-X459

Using a grid and m&m candies, we simulate the spread of disease. Students conduct the simulation and collect data to estimate parameters (in several ways) in a differential equation model for the spread of the disease.

1-014-DrainingContainers

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/H732-Q905

Given two rectangular circular cylinders of water with the same volume, but different radii, with a small bore hole of same radius on the center of the bottom through which water exits the cylinder, which empties faster?

1-013-SleuthingWithDifferentialEquations-ModelingScenario

26 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/M375-7X40

We present several situations in which differential equation models serve to aid in sleuthing and general investigations.

1-015-Torricelli-ModelingScenario

25 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/MVZ5-XY50

We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.

1-005B-ChemDataCollection-ModelingScenario

25 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/9WBX-VS55

A chemistry graduate student is collecting data on a chemical reaction and does a poor job of collecting the data. Indeed, he only collects the data at the start and end of a number of three minute intervals and does not note the times.

1-040-OutcomeSavings-ModelingScenario

25 May 2022 | Contributor(s): Brian Winkel | doi:10.25334/H3TH-E750

We ask students to build a model for a savings account and to determine the monthly deposit in a savings account in order meet a long term savings goal.