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We place here and in the Supporting Docs all the materials in support of the SIMIODE Remote Teaching Module - Modeling the Spread of Oil Slick.

This module contains

1) (Below and separate file in Supporting Docs) A brief Teaching Guide with an overview of the content and activity, including any necessary prerequisite material

2) Videos and materials to assign to students

3) An explanatory video for instructors on how to implement the module in an online format

We will offer a YouTube Version and SIMIODE Version in Supporting Docs. The YouTube version permits stop and start and bounce around while the SIMIODE version should be downloaded for full mobility.

4) (Included in the brief Teaching Guide below) An assessment guide

Additionally, we will be hosting live Q&A sessions in the future and posting FAQ guides.

For the Q&A session we will use ZOOM meeting software without recording, but we will post FAQ information based on Chat and meeting conversation. We will discuss the lesson experience and issues about teaching, interactions, and online experience.

This Teaching Module is about first order, linear, ordinary differential equations. The class lesson starts with analyzing data on the size of an oil slick in square miles taken from aerial photographs at irregular and unknown times.

The file Module-OilSlick is the file used for presentation for teachers of the Teaching Module while the file Module-Class-OilSlick is the actual lesson material suitable for use with class.

The formats for both are using LaTeX Beamer structure which permits production of PowerPoint like presentation using the mathematical typesetting capabilities of LaTeX. Both the original .tex file and the produced .pdf files are offered.

The class assignment is for the student to write up an informal flow of the modeling approach and to use the model to determine the actual times of the photographs since the photographer never really recorded these times, only photographs at several times and then ten minutes later in each of these observations.

Resources for this Remote Teaching Module are in the Supporting Docs with originating Modeling Scenario **1-005-S-OilSlick ** (Student Version) and **1-005-T-OilSlick** (Teacher Version)

**Teaching Guide for SIMIODE Remote Teaching Module OilSlick**

Prepared by Brian Winkel, Director SIMIODE

**Overview of Content**

S**ource: ** Modeling Scenario in SIMIODE at **www.simiode.or****g**.

- 1-005-S-OilSlick
**https://www.simiode.org/resources/196**- the Student Version of the material in which only the STATEMENT of the problem is offered. - 1-005-T-OilSlick
**https://www.simiode.org/resources/184**- the Teacher Version of the material in which the STATEMENT and COMMENTS by the author are presented discussion solution strategies and pedagogical issues.

The activity involves building a first order liner ordinary difference (leading to differential) equation from plots of differencing or average rate of change in the data and then solving the differential equation using separation of variables or integrating factor. The model is then used to interpret the data and fill in missing information about the data.

This is a modeling opportunity in which data is presented to the student in an incomplete manner, i.e., we have data on the size of an oil slick from aerial photographs taken at ten minute intervals, BUT the actual times at which the photographs were taken is NOT KNOWN. Therein lies the challenge. For students often want to plot (and we advocate) the data FIRST, but there is no way to plot Size of Slick Vs Time. So student struggle with how to get started, what to plot.

A complete description of many different student strategies (but by no means complete for students over the years and I have been using this material – some 40 years now(!)) is found in the Teacher Version Source above.

We offer a **teacher YouTube video** in which we present an introduction and the lesson offered to students. We also offer a **student YouTube video** in which we provide a ready to use lesson for students with assignment as well. We also offer versions of these videos in the Supporting Docs for this resource for immediate dlownload.

**Prerequisite Material**

- Differencing data to approximate derivative or rate of change over time interval
- Plotting data in some kind of software, e.g., spreadsheet or by hand
- Solving first order, linear ordinary differential equation using (a) separation of variable, (b) integrating factor, or (c) technology, e.g., software which student possess, SAGE, Maple, Mathematics, or Wolfram Alpha (
**https://www.wolframalpha.com/**).

This model could be used to introduce and motivate these prerequisite techniques as well.

**Nature of Activity**

Students use data to build a first order, linear differential equation; solve the differential equation; and use the solution to determine aspects from the incomplete data offered at the start of the activity.

**Videos and materials to assign to students**

We direct students to the actual statement of the problem in Source item above: 1-005-S-OilSlick **https://www.simiode.org/resources/196** - the Student Version of the material in which only the STATEMENT of the problem is offered.

We offer a pdf of the completed modeling activity from start to finish in the file Module-Class-OilSlick.pdf. The file is constructed using LaTeX Beamer and we also enclose the .tex file and all image files for customization, editing, and use by individual faculty.

We offer an Excel Spreadsheet file 1-005-T-Excel-OilSlick-TeacherVersion.xslx in which a complete analysis of the problem is rendered. We also offer an Excel Spreadsheet file 1-005-Excel-OilSlick-DataOnly.xslx file with the data only.

**Explanatory video for instructors on how to implement the module in an online format.**

In the file Module-OilSlick.pdf constructed using LaTeX Beamer we present slides in which we lead the faculty through a development of the modeling activity. We also offer a video walk through of this material with an introduction to the teacher, a middle section from the student perspective, and a closing section as if we were conducting a webinar for teachers to engage them and assist them in realizing the potential and use of this material.

**An assessment guide**

Each teacher has to decide how to use this material and hence how and what to assess. We offer here our Assignment (found in this material) we have traditionally used for this activity. Recall, we have been using this material successfully for over 40 years!

**Assignment**

- Write an overview of the modeling process to obtain s(t), the size of the oil slick in square miles, at time
*t*hr, using the data. - Use your model for
*s(t)*to plot size of the oil slick over time. - With your model ascertain the exact times at which the photographs were taken.
- Plot the model over the data to affirm or validate your model.
- Predict the long-term behavior of the oil slick.

This could be an in-class lab in which all the students gain the experience of the full modeling experience to completion and the teacher asks them to organize, communicate, and write-up their understanding of the work as a homework assignment. This is how I have used this in the past.

So we would spend about 30 minutes in class as I let them experience the initial frustration (in small groups) with being unable to plot the data in the traditional manner of size vs. time and coming up with an alternative plotting scheme, say, using differencing.

Question (1) is the heart of the write-up and I would assign 50% of the grade to this with attention to things like units, proper use of graphics, logical connection and flow, modeling assumption, etc.

Question 3 is a bit tedious as they have to solve the equation for *t* when setting their model in *t* equal to specific data points and this would take a bit of algebra (logging stuff as needed) or using a computer algebra solving command.Questions (4) and (5) demand use of a plotting routine and some observations of long-term behavior, either from the plot or from the form of the solution.

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