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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Slime Spread}





\markboth{Slime Spread}{Slime Spread}

\author{Brian Winkel,
Director, SIMIODE\\ 
Clhardon OH 44024USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We offer a video \cite{SlimeSpread2018} showing real time spread of a cylinder of slime and challenge students to build a mathematical model for this phenomenon.
\end{abstract}


\section*{SCENARIO DESCRIPTION}
Inspired by the article \cite{Single1999} we decided to see if we could model the spread of slime. 

We found a recipe (among oh so many!) for making slime on the Internet (see the Appendix below for details) and set out a ``cylinder'' of slime on a polar graph paper grid. In a video we made and posted on YouTube \cite{SlimeSpread2018} we  enable the viewer to see the radii of the slowly expanding circular ``puddle'' or ``glob'' of slime using a cookie cutter mold to hold the cylinder of slime in place until initial release of the slime disk.

Here, we offer  the opportunity to create  several models, truly empirical models based on shape and empirical models based on ``behavior," but none of these models are really based on first principles of physics. We further encourage differential equations models in which the rate of growth of the area of the ``glob'' is modeled.

We collected the data in a 65 degree Fahrenheit room using an iPad 4 and Time Lapse Video feature and made a video \cite{SlimeSpread2018}  of the phenomena which we posted on YouTube. We turned out a 1/4 cup (or 59.14 cm$^3$) measuring cup's worth of slime in the form of a cylinder and videoed its spread on a polar plot graph paper.  

We present a screen shot of the video \cite{SlimeSpread2018} in Figure~\ref{fig:SlimeSpreadImage}.

\begin{figure}[ht]
	\centering
	% Requires \usepackage{graphicx}
	\includegraphics[width=11 true cm]{SlimeSpreadImage}
	
	\parbox{8.5 true cm}{\caption{\label{fig:SlimeSpreadImage}Screen shot of paused video \cite{SlimeSpread2018} of spread of slime for collecting data with which to compare a modeling result.}}
\end{figure}

In the video the time in the lower left is on an iPhone. We read off the radius of the approximate circle on the tick marked axis on the polar grid lines to the right of the slime ``puddle."  The original amount of slime was approximately 1/4 cup or 59.14 cm$^3$.



Data Collected from the resulting mp4 movie SlimeSpread \cite{SlimeSpread2018}  which can be played  with most media player software was made with home made slime using a cookie cutter mold to hold until release of the slime disk. The data we collected is offered in Table~\ref{table:data} in the form of time in seconds (s) and area of the slime circle (estimated from the radius using the tick marked axis on the polar grid lines) in centimeters (cm$^2$).



\begin{table}[t]
\small
\begin{center}
\begin{tabular}{|r|r|}\hline 
Time (s)& Area (cm$^2$)\\ \hline\hline 
 0. & 32.1699 \\ \hline
 7.954 & 34.2119 \\ \hline
 23.884 & 36.3168 \\ \hline
 59.979 & 38.4845 \\ \hline
 95.973 & 40.715 \\ \hline
 144.016 & 43.0084 \\ \hline
 203.952 & 45.3646 \\ \hline
 295.954 & 47.7836 \\ \hline
 419.936 & 50.2655 \\ \hline
 591.95 & 52.8102 \\ \hline
 832.519 & 55.4177 \\ \hline
 1140.52 & 58.088 \\ \hline
 1459.48 & 60.8212 \\ \hline
 2258.49 & 63.6173 \\ \hline
 3406.46 & 66.4761 \\ \hline
\end{tabular}
\normalsize

\parbox{8.9 true cm}{\caption{\label{table:data} Data collected from the Spread of Slime video \cite{SlimeSpread2018}.}}
\end{center}
\end{table}

We offer a plot of the data in Figure~\ref{fig:SlimeDataPlot} to give the reader some idea of the process of the spread of slime. Notice how the growth slows down as time goes by.

\begin{figure}[h]
	\centering
	% Requires \usepackage{graphicx}
	\includegraphics[width=11 true cm]{SlimeDataPlot}
	
	\parbox{11.6 true cm}{\caption{\label{fig:SlimeDataPlot} Plot of the collected data  from the Spread of Slime video \cite{SlimeSpread2018}.}}
\end{figure}



\section*{APPENDIX - VIDEO DESCRIPTION}

We present here the description offered for the SlimeSpread Video on YouTube\cite{SlimeSpread2018}.

\vspace*{.4 true cm}

Here we present a video of the Spread of Slime for the collection of data in an attempt to model the spread of slime using a differential equation or empirical model.

\vspace*{.1 true cm}
We collected the data in a 65 degree Fahrenheit room using an iPad 4 and Tome Lapse Video feature. We turned out a 1/4 cup measuring cup's worth of slime in the form of a cylinder and videoed its spread on a polar plot graph paper whose length units are in cm.  
\vspace*{.1 true cm}

The time in the lower left is on an iPhone. We read off the radius of the approximate circle on the tick marked axis on the polar grid lines to the right of the slime "puddle."  
\vspace*{.1 true cm}

The original amount of slime was approximately 1/4 cup or 59.14 cm$^3$.
\vspace*{.1 true cm}

Here is a sample of the data we obtained in (time, and right hand radius read) format:
\vspace*{.1 true cm}

data = \{(1.959, 3.2), (9.913, 3.3), (25.843, 3.4), (61.938, 3.5), 
(97.932, 3.6), (145.975, 3.7), (205.911, 3.8), (297.913, 3.9), 
(421.895, 4.0), (593.909, 4.1), (834.478, 4.2), (1142.476, 4.3), 
(1461.437, 4.4), (2260.446, 4.5), (3408.416, 4.6)\}.
\vspace*{.1 true cm}

Time is measured in seconds and radius is measured in cm.
\vspace*{.1 true cm}

\subsection*{Recipe for slime used in this video}

\begin{itemize}
\item [] 4 oz Elmer's glue
\item [] 1 tsp laundry borax
\item [] 3/4 cup of water
\item [] food coloring
\end{itemize}

Dissolve the laundry borax in 1/4 cup of water. Combine the food coloring, glue, and 1/2 cup  of water in a bowl. Stir in the borax solution. The mixture will congeal immediately. Drain off any excess water and store slime in a sealed container. The slime will last for at least a week in the refrigerator, but will lose moisture with time and handling.  Actually in sealed jar it lasts for months!

\begin{thebibliography}{99}


\bibitem{Single1999} Single, R. 1999. Difference Quotients, Derivatives, and Data Modeling with Slime. {\it PRIMUS\/}.  9(3): 279-288.


\bibitem{SlimeSpread2018} SlimeSpread. 2018. Video of Spread of Slime.  \href{https://www.youtube.com/watch?v=Bvx1Et2Sg\_w}{https://www.youtube.com/watch?v=Bvx1Et2Sg\_w}. Accessed 19 May 2020.



\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

We suggest  two avenues, (1) an empirical approach in which students fit, based on shape, the data to a function form they believe might fit  as suggested by the data, and (2) an approach using a differential equation with reasonable assumptions to model the phenomena. Of course, neither  of these gets to basic physical principles. However, the differential equation model gives more plausibility to the model than just an empirical fit and causes the students to conjecture and follow through with plausible model.

In an accompanying Mathematica notebook file 1-053-T-Mma-SlimeSpread-TeacherVersion.nb and a pdf version for those who do not do Mathematica we attack this modeling problem in a number of ways to obtain the surface area of the ``puddle'' of slime, $A(t)$ in cm$^2$.
\begin{itemize}
\item First empirical model based on the shape of the data, $A(t) = \gamma\, t + \alpha \left(1 - e^{-\beta t}\right)$.  
\item Second empirical model based on the shape of the data, $A(t) = \gamma + \alpha \left(1 - e^{-\beta t}\right)$.   
\item Differential equation model,  $A'(t) = \alpha  V/A(t)$. 
\item Differential equation model,  $A'(t) = \alpha  V/A(t)^r$.
\item Differential equation model,  $A'(t) = \alpha e^{-\beta A(t)}$.
\item Differential equation model,  $A'(t) = k (ATV - A(t))$ where $ATV$ is apparent ``end''  or ``terminal'' area.
\item Differential equation model,  $A'(t) = k (ATV - A(t))^r$ where $ATV$ is apparent ``end''  or ``terminal'' area.
\item Differential equation model,  $A'(t) = k (ATV - A(t))^2$ where $ATV$ is apparent ``end''  or ``terminal'' area.
\item Differential equation model,  $A'(t) =  k (ATV - A(t)) {\rm Vol}_0/(A(t) *{\rm hVal})$ where Vol$_0$ is initial volume of cylinder of slime and hVal is the initial height of the cylinder of slime.
\item Differential equation model,    $A'(t) =  k (ATV - A(t))^2 {\rm Vol}_0/(A(t) *{\rm hVal})$ where Vol$_0$ is initial volume of cylinder of slime and hVal is the initial height of the cylinder of slime.
\end{itemize}



}





\end{document}
