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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\SPREAD OF OIL SLICK}


\markboth{Spread  of Oil Slick}{Spread  of Oil Slick}






\author{Brian Winkel,
Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode
}

\makeStitlePDFLaTex


\begin{abstract}
We describe a modeling activity for students in  which modeling with difference and differential equations is appropriate. We have used this model in our coursework for years and have found that it enlightens students as to the model building process and parameter estimation for a  linear, first-order, non-homogeneous, ordinary differential equation. The activity also provides a lesson in ``knowing what to do when you do not know what to do."\cite{Polya1945}  
\end{abstract}


\section*{SCENARIO DESCRIPTION}



An oil slick spreads at sea. From time to time, but irregularly, a helicopter is dispatched to photograph the oil slick. On each trip, the helicopter arrives over the slick, the pilot takes a picture, waits 10 minutes, takes another, and heads home. On each of seven trips the size (in area) of the slick is measured from both photographs.  


\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{Size of Slick (square miles)}\\ \hline \hline
Initial Observation & 10 min.~later \\ \hline
1.047 & 1.139 \\ \hline
2.005 & 2.087 \\ \hline
3.348 & 3.413 \\ \hline
5.719 & 5.765 \\ \hline
  7.273 & 7.304 \\ \hline
  8.410 & 8.426 \\ \hline
  9.117 & 9.127\\ \hline
\end{tabular}

\parbox{12.6 true cm}{\caption{\label{table:initialdata}  Data on spread of an oil slick. }}
\end{center}

\end{table}
\vspace*{-.75 true cm} \

\begin{itemize}
\item [(a)] Build a model for the size of the oil slick at time $t$. 
\vspace*{-.15 true cm} \item [(b)] Predict the future size of the oil slick, say at $t = 10$ min, $t = 20$ min, $t = 120$ min.
\vspace*{-.15 true cm} \item [(c)] Plot your model of the size of the oil slick as a function of time.
\vspace*{-.15 true cm} \item [(d)] Find the time at which the oil slick is 8 square miles. 
\vspace*{-.15 true cm} \item [(e)] Determine the time of each of the observations for the first, third, fifth, and seventh initial observations.
\end{itemize}

\teacher{

\section*{NOTES FOR TEACHERS}

We attach the essence of a complete article \cite{Winkel2010} about the use of this problem over many years in our teaching.  In this article we describe how we introduce this modeling scenario  to students and offer several student developed strategies for building a model and addressing the issues raised in the statement. We compare the various solution strategies.

\section*{Introduction\label{intro}}
We  share  one of our favorite problems, a problem which is rich in learning experiences and solution strategies. We believe this problem captures the essence of what we are and can be as teachers of mathematics while illustrating many technical, professional, and, at times, emotional aspects of our lives together as students and teachers.


The  problem (without solution) originally appeared in a 1975 article \cite{Taylor1975}   in a lovely pamphlet journal, {\it Applied Mathematics Notes de Math\'ematiques Appliqu\'ees\/}. The journal was a short-lived  publication of the Canadian Mathematical Congress and was meant to encourage the teaching of applied mathematics in undergraduate studies. In the more than 35 years since I first read this problem I have used this problem  in many courses. It never fails to captivate and challenge the students and it always gets them to move beyond where they are, to grow, and, to quote George Polya, to know ``what to do when you do not know what to do."\cite{Polya1945}


We ask the reader to consider this problem, doodle with it, give it an earnest try, and get somewhat frustrated ---all the better to enjoy our journey.  To get the most out of this article you should take at least 15 minutes to do this right now, before reading  the strategies for attacking the problem. 


\begin{quote}
An oil slick spreads at sea. From time to time, but irregularly, a helicopter is dispatched to photograph the oil slick. On each trip, the helicopter arrives over the slick, the pilot takes a picture, waits 10 minutes, takes another, and heads home. On each of seven trips the size (in area) of the slick is measured from both photographs. Build a model for the growth of the oil slick.\cite[pp.~73-74]{Taylor1975} 

  

\begin{table}\label{table:initialdata}
\begin{center}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{Size of Slick (square miles)}\\ \hline \hline
Initial Observation & 10 min. later \\ \hline
1.047 & 1.139 \\ \hline
2.005 & 2.087 \\ \hline
3.348 & 3.413 \\ \hline
5.719 & 5.765 \\ \hline
  7.273 & 7.304 \\ \hline
  8.410 & 8.426 \\ \hline
  9.117 & 9.127\\ \hline
\end{tabular}

\parbox{12.6 true cm}{\caption{\label{table:initialdata}  Data on spread of an oil slick. \cite[p.~74]{Taylor1975}}}
\end{center}
 \vspace*{-.4 true cm}

\end{table}
 
\end{quote}


In a most recent reincarnation my wording, which accompanies the data  in Table~\ref{table:initialdata}, to the students went something like this:

\begin{quote}
An oil slick spreads at sea. From time to time, but irregularly, a helicopter is dispatched to photograph the oil slick. On each trip, the helicopter arrives over the slick, the pilot takes a picture, waits 10 minutes, takes another, and heads home. On each of seven trips the size (in area) of the slick is measured from both photographs. See Table~\ref{table:initialdata} for the data.
\begin{itemize}
\item [(a)] Build a model for the growth of the oil slick at time $t$. 
\item [(b)] Predict the future size of the oil slick, say at $t = 10$ min, $t = 20$ min, $t = 120$ min.
\item [(c)] Plot your model of the size of the oil slick as a function of time.
\item [(d)] Find the time at which the oil slick is 8 square miles. 
\item [(e)] Determine the time of each of the observations for the first, third, fifth, and seventh initial observations.
\end{itemize}
\end{quote}

As you can see, I have added some questions {\it after\/} students have obtained a model to show them they can use the model to do some good things, such as  telling the time at which these observations actually took place. I am a bit more instructive and specific in my questions. This  gives students rungs on which to place their feet as the climb this ladder of a problem to success.



\section*{The Frustration of Modeling\label{section:Frustration}}

My  intention is to offer the activity as a challenging (we shall see why, if you have not already discovered that from our admonition to try it) problem  to open students' eyes to differencing data in order to build a difference or differential equation model. However, the problem can be presented to students who have not even had calculus, but who do know and understand the concept of slope  in terms of rate of change, say, of  the size of slick in square miles per ten minute interval.

Since I first used the problem in a differential equations course and pushed the students with the usual flurry of hints during class I always expect (and always receive) solutions involving the construction of a differential equation, estimates of the parameters in the equation,  the solution of said differential equation, and its use in ascertaining times of events.  I usually assign this problem over a week's time period while doing other  problems in a first course in differential equations.

One year I had a very good student who thought outside  the box. As I read the submitted solutions I came to his. It  did not even have a differential equation. It was poorly written, not explaining the steps, and made preposterous claims such as ``See, I did not even need a differential equation to solve it." What was he doing? I had to dig in and I had to do it quickly, for I usually returned papers by the next class meeting, during which we discussed the broader issues  and the techniques used. We will come to this different approach than I had envisioned in later sections of this scenario. 

   

The  students  working on this problem ALWAYS meet with a good bit of frustration. The discussions we present usually take place before the students bring the problem home for further work. Discussion also  takes place as an after action report or mid-course correction for some, in which we process what is happening to each of us as we battle the problem.  

Consider the problem again. It involves data from observations. What do we always tell students to do when presented with data?  Plot the data  in order to get a picture so you might have a sense of what you have before you. This command, ``Thou shalt plot," causes the main frustration, for we {\it can\/} plot at $t = 0$ the size $s(0) = 1.047$  and we can then plot the size 10 minutes later as $s(10) = 1.139$, where $s(t)$ is the size of the oil slick in square miles  $t$ minutes after the first observation.  Here comes  the frustration. What do we plot next?  We do not know when the next observation took place so we have no independent time to which we can associate the size 2.005. Some students are quick to point out, ``Yes, but if we could get that time, say $t_1$, then we would know $s(t_1 + 10) = 2.087$." Others chime (chide?) in, ``But we do not know $t_1$."

Now begins a series of false assumptions and rationalizations. One student offers, ``Let's assume the length of time intervals between the second observation and the next first observation are always the same for each pair of observations."  She is reproached by classmates. To which she says, ``Well, if we do not know that time interval how are we going to plot the next point after $s(10) = 1.139?$" Discussions about what assumptions really mean in a modeling problem ensue and assuming something about the times between each of the ten minute observation intervals just does not go down well with all the students. Thus we abandon this strategy.  Now, we face an even bigger frustration, ``Which way do we go?" Guess who speaks up?  You got it!!! George Polya, ``knowing what to do when you do not know what to do."\cite{Polya1945} That does not help us actually find what to do, but it reassures us that it is fine to be in this predicament. The teacher is quiet. The room is silent.

\subsection*{We could be on to something here}

Someone usually says (and on VERY rare occasion it is the teacher, for eventually students discover this idea), ``Well, we do know how fast the oil slick size is changing in a bunch of intervals. Maybe we could compute these."  Quite often we present this exercise in an Excel spreadsheet. This  offers a visual stimulus and prompts responses because of the way spreadsheets present the data. Our spreadsheet data may look something like Table~{\ref{table:initialdata}.
  
\begin{table}\label{table:bunchdata}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
\multicolumn{2}{|c|}{Size of Slick (square miles)}& Change in Size of Slick \\ \hline \hline
Initial Observation & 10 min. later & 10 min. change\\ \hline
1.047 & 1.139 & 0.0092\\ \hline
2.005 & 2.087 & 0.0082\\ \hline
3.348 & 3.413 & 0.0065\\ \hline
5.719 & 5.765 & 0.0046\\ \hline
  7.273 & 7.304 & 0.0031\\ \hline
  8.410 & 8.426 & 0.016\\ \hline
  9.117 & 9.127 & 0.0010\\ \hline
\end{tabular}

\parbox{12.6 true cm}{\caption{\label{table:bunchdata} Data on spread of an oil slick given in  \cite[p.~74]{Taylor1975} with computed change in size of oil slick in preparation for building a difference/difference equation model.}}





\end{center}
 \vspace*{-.6 true cm}

\end{table}




This is what will happen next.  It always does; never fails to happen. Someone will say, ``Yeah, but what good did that do because we still can't plot the change versus time." Silence.  This part is where the rush for the teacher comes. Some small voice suggests (and it is not always the good student, but  often it is the very timid one in the back  who says in an interrogatory sentence, not a declarative one) ``Could we plot the change vs. size?" As the teacher you know they are on to it. They have taken a major step toward creating this differential equation you have been hoping they would build. Patience is still the order of the day. ``What good would that do?" says one critical (and perhaps still frustrated) student.  ``We still do not know at what times these changes took place at." If necessary the teacher can say something like, ``Well, we could just plot the change vs. size and see what we get. That at least is possible." Remember, we are in a spreadsheet with columns of data. Students are self-prompted to plot one column versus another. Why?  Well, because as George Mallory said of Mount Everest, ``Because it is there."\cite{Archer2009} So they plot and get Figure~\ref{fig:1-5-OilSlickDifferenceVsSizePlot}.



``Wow!" That is the usual response. We pause to reflect - we do a great deal of that in my courses. ``What just happened?" I say to them. Often they say we just ``tried something." They tried something special though. They took differences and so they took a step toward building a difference/differential equation. For most, when they see that kind of data they fit a line through it using Excel's TrendLine capability. In Figure~\ref{fig:1-5-OilSlickDifferenceVsSizePlot} we offer a {\it Mathematica\/} graphic. Sometimes we do our modeling in {\it Mathematica\/}, although {\it Mathematica\/} does not render the immediate juxtaposition of the columns and hence the visual hint on differencing and plotting as Excel does, thus prompting them to difference and plot.

\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{1-5-OilSlickDifferenceVsSizePlot.eps}

%\vspace*{-1.7 true cm}

\parbox{12 true cm}{\caption{\label{fig:1-5-OilSlickDifferenceVsSizePlot} The first post-frustration plot students render when they shift off plotting something against time and try to plot the change in size vs. size.}}

\end{figure}


\begin{figure}[h]
\centering
\includegraphics[width=10cm]{1-5-OilSlickLineOverDifferenceVsSizePlot.eps}

%\vspace*{-1.7 true cm}

\parbox{10.6 true cm}{\caption{\label{fig:1-5-OilSlickLineOverDifferenceVsSizePlot} The plot of rate of change of size of oil slick vs. size of oil slick with the best fit line superimposed upon the data.}}

\end{figure}

Excel's TrendLine will give the equation of this line or one could apply {\it Mathematica\/}'s {\tt Fit} command, {\tt Fit[data,\{1,x\},x]}, (or other software's regression command) to the data set of (change in oil slick, size of oil slick) pairs. The resulting linear equation, \vspace*{-.3 true cm}

\begin{equation}
y = -0.00100087 x + 0.0101644\,  ,
\end{equation}

\noindent is the best fit equation and the way we write a linear function. However, $x$ and $y$ are not the variables here. What are the variables? They are $x = s(t)$ and $y = \frac{\Delta s}{\Delta t}$, where $s(t)$ is the size of the oil slick at time $t$ and $\frac{\Delta s}{\Delta t}$ is the average rate of change of the oil slick over an interval of size $\Delta t$ at time $t$.  Thus, we have  constructed the following  relationship between change in size and size:

\begin{equation}
\frac{\Delta s(t)}{\Delta t} = -0.00100087\cdot s(t) + 0.0101644\,   , 
\end{equation}
where $\Delta t = 10$. See Figure~\ref{fig:1-5-OilSlickLineOverDifferenceVsSizePlot}.




\subsection*{We have a differential equation model}

We could branch to a difference equation approach or we could use this  average rate of change to give another approximation - the instantaneous rate of change or the derivative:
\begin{eqnarray}
\frac{ds(t)}{dt} = -0.00100087\cdot s(t) + 0.0101644\,   \label{eq:OilSlickDE}
\end{eqnarray}
where we recognize that $=$ could just as easily, and more properly, be replaced by  $\sim$, a close approximation, however.

Of course, the students are quick to point out that if we are going to solve this  differential equation we will need the initial condition found in our data, $s(0) = 1.047$. They are one step ahead of themselves now as they are out from underneath the previous frustrations and on a roll with respect to our study of   differential equations. A comfort zone has been reached as we now have a differential equation with initial condition  and our natural reaction is to solve it. We can separate the variables, we can use an Integrating Factor, or, as I do in my teaching, we can use {\tt DSolve} in {\it Mathematica\/}. Indeed, if we use this modeling activity before we actually discuss a differential equation or any solution strategies we have a reason, a motivation, to actually study  differential equations --- modeling first differential equations!

Before solving I ask them, ``What will we do with our solution once we get it?" Silence, but not for long, as someone always says, ``We will plot the solution and that will be our model."  I wonder aloud as to how we will check the goodness of the  model, to which they say they will see how it fits the data - frustration is in order, for we do not have a plot of the data. Silence.  Another (often) small voice says, ``We could find out what the $t$ values for each data point are by setting our solution $s(t)$ equal to 2.005, then 3.348, then 5.719, etc. and solving for $t$." Someone else chimes in, "Yeah.  We could then find what the size of the slick is 10 minutes after it is 2.005." Another adds, ``And it better be about 2.087." So they are off to do battle and to complete their write-up of this problem with some confidence that they learned how to build a model and, most importantly, that they understood the meaning of ``knowing what to do when you do not know what to do."\cite{Polya1945} I always give this as an individual write-up assignment, but I encourage discussion inside and outside of class. I  ask that they credit folks whose ideas they use.

\begin{table}\label{table:finaldata}
\begin{center}

\begin{tabular}{|c|c||c|c|} \hline 
\multicolumn{2}{|c|}{Size of Slick (square miles)}& \multicolumn{2}{|c|}{Times for Sizes (min)}\\ \hline \hline
Initial Observation & 10 min. later &Time Start&Time End\\ \hline
1.047 & 1.139 &0&10\\ \hline
2.005 & 2.087 &111.031&121.031\\ \hline
3.348 & 3.413 &290.926&300.926\\ \hline
5.719 & 5.765 &718.704&728.704\\ \hline
  7.273 & 7.304 &1149.52&1159.52\\ \hline
  8.410 & 8.426 &1650.68&1660.68\\ \hline
  9.117 & 9.127 &2169.45&2179.45\\ \hline
\end{tabular}


\parbox{12.6 true cm}{\caption{\label{table:finaldata} Data on spread of an oil slick given in  \cite[p.~74]{Taylor1975} with model predicted times from our differential equation model. The comparisons are quite good.}}
\end{center}
\vspace*{-.7 true cm}
\end{table}







\subsection*{Now let us put our solution to work for us}

We show students' results  and how there is a tremendous sense of satisfaction in fitting it all together. They determine the solution to the differential equation~(\ref{eq:OilSlickDE}):\vspace*{-.3 true cm}

\begin{eqnarray}
s(t) = 10.1556 -9.10862 e^{-0.00100087 t} \, .  \label{eq:OilSlickDESol}
\end{eqnarray}

We use this to determine the time for every observation (see Table~\ref{table:finaldata}), to plot (see Figure~\ref{fig:1-5-OilSlickDESolutionOverData}) the newly timed data points over the model solution, and to just sit back and say, ``We are good!" The teacher in me requests some reflection on our process and the students relive the frustration, the ``Why not difference the data?" moment, the ``What can we now plot against what?" question, the ``Wow!" reaction to the plot, and the ``Hey, I can solve this DE," reaction. It is all there. Modeling, analysis, moving beyond what we are, the ``knowing what to do when you do not know what to do."\cite{Polya1945}  We continually refer to Polya's quote when faced with difficult times and we make these into opportunities rather than frustrations. We have not quite climbed Mount Everest (but neither did Mallory, or did he? See \cite{Archer2009}.) but we have conquered our own internal inertia when faced with a difficult moment in our modeling process. It is a good feeling. The students and I savor it.  They remember it. I have strong evidence of the latter as this problem comes up in discussions with students years later when they visit or we reminisce.

\begin{figure}[h]

\centering
\includegraphics[width=10cm]{1-5-OilSlickDESolutionOverData.eps}

%\vspace*{-1.7 true cm}

\parbox{12.6 true cm}{\caption{\label{fig:1-5-OilSlickDESolutionOverData} A final, convincing plot of the differential equation model obtained from differencing our data vs.~the data itself, showing the specific times at which the size of the oil slick was given initially in each observation interval and  10 minutes later in each case. Two points for each set of observations are not really noticeable as the 10 minute separation is lost over the time interval, $[0,2500]$ minutes. }}
\vspace*{-.4 true cm}
\end{figure}
We set our $s(t)$ from (\ref{eq:OilSlickDESol}) equal to the values for the size of the oil slick in our original data to find the times at which the oil slick assumed these values.  We summarize this information in   Figure~\ref{fig:1-5-OilSlickDESolutionOverData}.







\section*{Something slightly different}


This strategy emerged  years ago when I was teaching in an engineering school (Rose-Hulman Institue of Technology)  after teaching and using this problem in a liberal arts college (Albion College) for 10 years, both with good students. This approach is a slight variation, nevertheless a genuine variation,  of our first modeling solution offered above. In this solution the work  leads to a differential equation whose solution, again, will give the size of the slick as a function of time. 

In class, while working on this problem in small groups, one student simply said, ``Why not plot the start vs. the finish over each observation period and see what we get?"   This is a direct result of the earlier frustration, i.e. of not having anything to plot vs.~time.     So we plotted the size of the oil slick after 10 minute vs.~original size in THAT interval and we show this plot  in Figure~\ref{fig:1-5-OilSlickLineInitialVsLater}. 


\begin{figure}[h]
\centering
\includegraphics[width=10cm]{1-5-OilSlickLineInitialVsLater.eps}

%\vspace*{-1.7 true cm}

\parbox{12.6 true cm}{\caption{\label{fig:1-5-OilSlickLineInitialVsLater} We present a plot of the 10 minute later observation vs.~the initial observation for each of the observation periods. }}

\end{figure}


The plot looks good and we ought to be able to do something with it.  From it we build a nice difference equation  \vspace*{-.3 true cm}
\begin{eqnarray}
s(t+10) = 0.989991*\cdot s(t) + 0.101644 \label{eq:differenceEquation}
\end{eqnarray}
after fitting a line to the data.

If we subtract  $s(t)$ from both sides  we can convert (\ref{eq:differenceEquation}) into a differential equation in  $s(t)$, the size of the oil slick at time $t$. Then by approximating $\frac{s(t+10) - s(t)}{10}$ with $\frac{ds(t)}{dt}$ we are in the same position as in the previous approach and obtain almost the same results as we did before.

Incidentally, {\it Mathematica\/} has a very nice command for solving difference equations:
\vspace*{.15 true cm}

{\tt RSolve[\{s[n + 1] == 0.101644 + 0.989991 s[n], s[0] == 1.047\}, s[n], 
   n]}

\vspace*{.15 true cm}

\noindent where $n+1$ refers to the $(n+1)^{\rm st}$ 10 minute interval and $s(n)$ is the size of the oil slick in square miles at the $n^{\rm th}$ 10 minute interval. {\tt RSolve} gives us a solution of the form:
 \vspace*{-.1 true cm}
\begin{eqnarray}
s(n) = 10.1553\cdot 0.989991^n\cdot 2.71828^{0.0100594
   n}-9.10826\cdot 0.989991^n
\end{eqnarray}
 \vspace*{-.5 true cm}

If we set $s(n) = 8.410$, the size of the slick at the start of the 6$^{\rm th}$ interval, we can use {\it Mathematica\/}'s workhorse command, {\tt FindRoot},  to see that $n = 164.252$. This means that at time $164.252\times 10 = 1642.52$ minutes after our first observation (we have 164.252 of these units of 10 minute intervals) the size of the slick was observed to be 8.410 square miles according to our new difference equation model. This is quite close to the value of 1650.68 found in the corresponding cell of Table~\ref{table:finaldata} where we used a  differential equation model. We point out to students that difference and differential equations are approximations of each other and in this case the difference equation, which is actually a form of Euler's method for solving the differential equation, renders values that are less (but not by much!) than those offered by the differential equation. \vspace*{-.3 true cm}




\section*{And now for something completely different\cite{MontyPython1971}}

Let us get back to the frustration of not having any points to plot beyond the first two observations, i.e., $(0,s(0))$ and $(10,s(10))$.  We just do not know a $t$ value for when the oil slick is 2.005 square miles which occurs at the beginning of the second set of observations. How could we find out when this occurs? That is a question some students asked several years back. After wrestling with the situation one student noted that if we know the rate at which the size of the oil slick increases in the first interval, i.e. $\frac{s(10) - s(0)}{10}$, this rate is a good estimate of the slope of the curve we seek eventually, i.e. the slope of the mystery function, $s(t)$.

Here it comes again. The students blow you away. They show you that they really  understand the meaning of a concept like slope in their usage of it to solve problems. You feel good about our system! So one student says to herself while working on this problem in a cranny in the olde library, ``If I just start at the point $(10,s(10))$ and head out in a direction of the slope $\frac{s(10) - s(0)}{10}$ I will eventually hit an  $s$  value of  2.005 and I can solve to determine WHEN I arrive at this $s$ value, i.e. what $t$ value will give rise to this newly obtained $s(t)$ value of 2.005? I can use my technology to do this." Let's go along with this student's approach.

First we need to find out the time of the initial observation of the second pass, i.e., $s = 2.005$, took place. We compute the slope or rate of change in the oil slick over the first observation time interval $[0, 10]$,

$$\frac{s(10) - s(0)}{10} = \frac{1.139-1.047}{10} = 0.0092\, .$$ 
Now starting at our last observation, $s(10)$ and heading out in the direction of the slope 0.0092 just when will we get to an $s$ value of 2.005? Form the equation of the line through $s(10) = 1.047$ with slope 0.0092 and find when it hits a value of 2.005. In {\it Mathematica\/} she does this quite easily with the standard point-slope method for the equation of the line \vspace*{-.4 true cm}

\begin{eqnarray}
{\tt Solve[(s - 1.139)/(t - 10) == (1.139 - 1.047)/10, s] // Simplify} \label{eq:line}
\end{eqnarray}
\noindent to obtain $s = 1.047 t + 0.955$. Setting $s = 2.005$ yields a $t$ value of 104.13 min.  ``So that is the time we make our second initial observation.  Simply amazing.'' she says to herself. ``This stuff works!''

Solving $s = 1.047 t + 0.955$ for all the oil slick sizes produces values of $t$ for each corresponding oil slick size in this ``something different" approach. However, one will find that it  would appear that we have calculated our between observation time intervals incorrectly. In particular, we have used a bigger than we need growth rate over the entire interval between  observations.  We need to estimate the growth rate in the interval between observation intervals and so we take the average of the growth rates from consecutive intervals (surrounding the desired time interval between pairs of observations) to determine the time it takes between observation intervals, i.e.  \vspace*{-.4 true cm}

\large

\begin{eqnarray}
\frac{\frac{(s(t_i+10) - s(t_i))}{10} + \frac{(s(t_{i-1}+10) - s(t_{i-1}))}{10}}{2}\, , \label{eq:slope}
\end{eqnarray}
\normalsize

\noindent  where $t_i$ is the time of the initial $i^{\rm th}$ observation. For each large time interval between each 10 minute period of observation intervals we use the computed slope from (\ref{eq:slope}) in  the right hand side slope information found in (\ref{eq:line}). Of course we also replace the 1.139 on the left hand side of  (\ref{eq:line}) with the appropriate 10 minute later size of the oil slick in each observation interval.

We present the answers in an  extension of  Table~\ref{table:finaldata} in Table~\ref{table:comparisondata}.  We note how close these numbers from our ``something different" approach are to the previous method using the differential equation model. Of course, our times between intervals  are slightly different as well.



\begin{table}[h]\label{table:comparisondata}
\begin{center}

\begin{tabular}{|c|c||c|c|c|c|} \hline 
\multicolumn{4}{|c|}{Differential Equation Model Approach} & \multicolumn{2}{|c|}{Something Different Approach} \\ \hline \hline
\multicolumn{2}{|c|}{Size of Slick (square miles)}& \multicolumn{2}{|c|}{Times for Sizes (min)}& \multicolumn{2}{|c|}{Times for Sizes (min)}\\ \hline \hline
Initial Observation & 10 min. later &Time Start&Time End&\,\,\, Time Start\,\, &Size of Slick \\ \hline
1.047 & 1.139 &0&10&0&1.047\\ \hline
2.005 & 2.087 &111.031&121.031& 109.54&2.005\\ \hline
3.348 & 3.413 &290.926&300.926&291.10&3.348\\ \hline
5.719 & 5.765 &718.704&728.704&716.60&5.719\\ \hline
  7.273 & 7.304 &1149.52&1159.52&1118.29&7.273\\ \hline
  8.410 & 8.426 &1650.68&1660.68&1598.93&8.410\\ \hline
  9.117 & 9.127 &2169.45&2179.45&2140.47&9.117\\ \hline
\end{tabular}

\parbox{12.6 true cm}{\caption{\label{table:comparisondata} Data on spread of an oil slick given in  \cite[p.~74]{Taylor1975} with model predicted times from our slope and move forward approach - something different.}}



\end{center}
 \vspace*{-.4 true cm}
\end{table}




\section*{A final approach of fitting a  function through the data}


Here again, we presume that at time $t_0=0$ we have an initial observation of 1.047 for the size of the slick and then at $t_0 + 10 = 10$ we have an observation of 1.139. Define the  times for the initial observation of each of the next 6 observation intervals  as $t_1$, $t_2$, $t_3$, $t_4$, $t_5$, and $t_6$. 

Because of the notion of saturation, i.e. the oil slick (if contiguous), even if only one molecule thick, will have a limiting size, we believe ONE reasonable model for the size of the oil slick, $s(t)$, might be
\begin{eqnarray}\label{eq:newmodel}
s(t) = A (1 - e^{-k\cdot t})\, ,
\end{eqnarray}

\noindent as this is a limiting or saturation model often found in the literature, for no one believes the oil slick will grow to an unlimited size. Indeed, there is often discussion that even if the oil slick is one molecule thick and contiguous there will be a finite limit to the area. Our task is to determine the parameters $t_1$, $t_2$, $t_3$, $t_4$, $t_5$, and $t_6$, as well as $A$, and $k$, where we have observations of the size of the slick at the start of each 10 minute observation at $t_i$ for  $i = 1, 2, \ldots, 6$ and the size of the slick at the end of each 10 minute observation at $t_i+10$ $i = 1, 2, \ldots, 6$ to be $s(t_i+10)$. We perform one ``giant" least squares, (\ref{eq:SSE}), between the observed data and our model in (\ref{eq:newmodel}) for the parameter set, $t_1$, $t_2$, $t_3$, $t_4$, $t_5$, $t_6$, $A$, and $k$  and minimize to find the best fitting parameters and then see how good our determined parameter set, when in place for our function, actually predicts the observations. Here we go. We get {\it Mathematica\/} to {\tt FindMinimum} for the  expression in (\ref{eq:SSE}).

\vspace*{-.5 true cm}
\begin{eqnarray}
SSE(t_1,t_2,t_3,t_4,t_5,t_6,A, k) = \sum_{i=1}^6 \large(A \large(1 - e^{-k\cdot t_i}\large)- \hat{s}(t_i)\large)^2 + \large(A \large(1 - e^{-k\cdot (t_i+10)}\large)- \hat{s}(t_i+10)\large)^2 \, , \label{eq:SSE}
\end{eqnarray}
\noindent and here is what {\it Mathematica\/} returns:

\begin{eqnarray*}
&&t_1 = 111.306,\,\,\, t_2 = 291.196,\,\,\,  t_3 = 719.367,\,\,\,  t_4 = 1149.13, \\
  &&t_5 = 1645.6,\,\,\,  t_6 = 2156.12,\,\,\,  A = 9.13783,\,\,\,  k= 0.000995548\, .
\end{eqnarray*}

\noindent  $\hat{s}(t_i)$ represents the $i^{\rm th}$ initial observation value for the size of the oil slick. With these time values we can plot when all of our observations took place and compare (see Figure~\ref{fig:1-5-OilSlickCurveFit}) this to the curve of our model
$$s(t) = 9.13783 \left(1 - e^{-0.000995548\cdot t}\right)\, .$$



\begin{figure}[h]
\centering
\includegraphics[width=10cm]{1-5-OilSlickCurveFit.eps}

%\vspace*{-1.7 true cm}

\parbox{12.6 true cm}{\caption{\label{fig:1-5-OilSlickCurveFit} A final, convincing plot of the curve fitting approach to modeling the spread of an oil slick in which we find the parameters $A$ and $k$ in the saturation model $s(t) = A (1 - e^{-k*t})$.  From this model we can compute the times at which the oil slick takes on  the observed values.  We plot our model through the original observation data with our computed times for these observations from our model.}}

\end{figure}




\section*{CONCLUSION}



We have offered up several  approaches to a modeling opportunity and described how we use this problem in a number of settings. We have marveled (with you, hopefully) how we get many different and ingenious student responses.

Coming to the crucial and frustrating point of not knowing what to do with regard to attempting to plot data brings students to a moment when they will need to improvise and to try several different paths.  This is a great learning opportunity in the art of modeling and problem solving. Faculty should create and seize such moments! We have taken the time to carefully demonstrate the results of this opportunity with this problem which has ``worked" in so many ways for us and our students, as outlined above. Indeed, this problem is a wonderful illustration of Polya's adage that problem solving is ``knowing what to do when you do not know what to do."\cite{Polya1945}



This write-up has been about a  rich teaching experience the author has had over many years and the attendant learning experiences his students have demonstrated.  We have shown the variety of student responses and the richness of the approaches as examples of how modeling activities in a differential equations course can offer many tracks to success and can introduce, motivate, and sustain the study of differential equations. 

I suggest you try this rich problem in one of your courses you deem appropriate. You have seen some of the things my students did. See how your students will amaze you with their personal perspectives and approaches. It has been worth the journey EVERY time I have used this problem ever since I found it 35 years ago and it NEVER fails!! 



\begin{thebibliography}{99}

\bibitem{Archer2009} Archer, J. 2009. {\it Paths of Glory\/}.  New York:  St. Martin's Paperbacks.

\bibitem{MontyPython1971} Monty Python.  1971. {\it And now for something completely different\/}. YouTube.  \url{https://www.youtube.com/watch?v=K2P86C-1x3o}. Accessed 25 April 2015.

\bibitem{Polya1945} Polya, G. 1945. {\it How to Solve It\/}. Princeton NJ: Princeton University Press.


\bibitem{Taylor1975} Taylor, P. D. 1975. Calculus with Modelling for Science Majors. {\it Applied Mathematics Notes\/}. 1(2): 64-74. 

\bibitem{Winkel2010} Winkel, B. J. 2010. Modeling the Spread of an Oil Slick  with Irregular Information. {\it International Journal of Mathematical
Education in Science and Technology\/}. 41(8): 1085-1096.

\end{thebibliography}

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