We place here and in the Supporting Docs all the materials in support of the SIMIODE Remote Teaching Module - Modeling the Falling Column of Water.
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 collecting data on a falling column of water from a YouTube video and building a model (empirical and analytic) by first deriving Torricelli's Law from some first principles and then applying this law. This work involves first order, nonlinear, ordinary differential equations.
The file Module-Torricelli is the file used for presentation for teachers of the Teaching Module while the file Module-Class-Torricelli 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.
Teaching Guide for SIMIODE Remote Teaching Module Torricelli
Prepared by Brian Winkel, Director SIMIODE
Overview of Content
- 1-015-S-Torricelli https://www.simiode.org/resources/488 - the Student Version of the material in which only the STATEMENT of the problem is offered.
- 1-015-T-Torricelli https://www.simiode.org/resources/463 - 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 collecting data from a YouTube video on a falling column of water and building first an empirical model and then an analytic model based on Torricelli's Law derived in this material from first principles, in particular the Law of Conservation of Energy.
- Some familiarity with spreadsheet software and plotting of data and empirical modeling.
- Some understanding of Conservation of Energy Law, but it is all introduced, defined, developed, and applied in the material here.
- Solving first order, linear ordinary differential equation using separation of variable or technology, e.g., software which student possess, SAGE, Maple, Mathematics, or Wolfram Alpha (https://www.wolframalpha.com/).
- Constructing sum of square error function in terms of a parameter between model and actual data which is then minimized to determine best fitting parameter.
This model could be used to introduce and motivate the solution prerequisite technique as well.
Nature of Activity
Students use data to build a first order, nonlinear differential equation; solve the differential equation; and use the solution to determine a parameter related to the size of the exit hole opening for the water at the base of the column of water.
Videos and materials to assign to students
We direct students to the actual statement of the problem in Source item above: 1-015-S-Torricelli https://www.simiode.org/resources/488 - 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-Torricelli.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 CompleteModel11Over64InchHole.xlsx in which a complete analysis of the problem. We also offer an Excel Spreadsheet file Data11Over64InchHole.xlsx with the data only.
Explanatory video for instructors on how to implement the module in an online format.
In the file Module-Torricelli.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. We offer a student version video which teachers can use as is with their class.
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!
Write an overview of the modeling process to obtain h'(t) using first principles - not all details, just highlights. Arrive at the model.
Collect data for your team's cylinder from https://www.simiode.org/resources/488.What if we took many data points? Few data points? Try both by taking a subset of ALL the data points you took for ``few data'' points and see how your parameter b fares.
- In Excel we do these steps to determine best fit parameter b:
- Compute our model value of the height h(ti) at time ti.
- Take the differences between actual data and model prediction, i.e. hi - h(ti), and square these differences, (hi - h(ti))^2.
- Sum these square errors to obtain SSE(b).
- Use Excel's Solver to minimize SSE(b).
- Read the value of b and put it in our model as best parameter estimate of b.
- Plot our best model values on the same axes as our data and compare.
- Collect the parameters b for the various outflow hole sizes from different videos selected by teams and see if there is any relationship between hole size and b
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 developing the model, both empirical and with much more time on analytic model from Conservation of Energy Law.
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 (2) asks them to collect data.
Question (3) is where the students do their analysis with the cylinder of their choice. This is worth 25%. of the grade.
Questions (4) and (5) demand determination of students' individual parameter, b, and further the collection and analysis of the differnet b values for the different cylinder configurations. For the collaboration necessary and the observations and reporting of different values of the several configuration's b values I would assign the final 25% of the grade.
Cite this work
Researchers should cite this work as follows: