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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\ANT TUNNEL BUILDING}



\markboth{Ant Tunnel Building}{Ant Tunnel Building}






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


\makeStitlePDFLaTex


\begin{abstract}
 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. 
\end{abstract}

\section*{SCENARIO DESCRIPTION}




How long does it take an ant to build a tunnel? That seems like a reasonable question. If you ever had an ant colony purchased by a well-meaning aunt for you in grade school you may have watched ants building industriously and you just might have an idea on this.  To answer the question we  might need some narrowing of scope, some simplification, and certainly some identification of terms and variables  before we can get a nice answer. Let us identify some variables and then together make some assumptions which will lead to a mathematical model.

\vspace*{.2 true cm}
Let $x$ be the length of the tunnel in feet that an ant builds.

\vspace*{.2 true cm}
Let $T(x)$ be the time in hours it takes the ant to build the tunnel of length $x$. 

\vspace*{.2 true cm}
We can get some idea of our situation by making a sketch in Figure~\ref{fig:1-007-ModelingOneAnt1}.


\begin{figure}[!h]
\begin{center}
\includegraphics[width=10.9cm]{1-007-ModelingOneAnt1.eps}



\vspace*{-11.5 true cm}
\parbox{12.2 true cm}{\caption{\label{fig:1-007-ModelingOneAnt1} Crude drawing for ant tunnel building model. $x$ is the length of the tunnel and $T(x)$ is the time it takes an ant to build a tunnel of length $x$.}}
\end{center}

\end{figure}
\begin{itemize}


\item [a)] Write down several candidate functions for $T(x)$ and give one or two statements in each's defense and one or two statements against each.


\end{itemize}

We see that attempting to jump right on top of $T(x)$ can be hard.  So, instead of going after $T(x)$ directly let us examine Figure~\ref{fig:1-007-ModelingOneAnt2}.  Consider some assumptions which (i) reflect the reality of such a situation and (ii) might make the model simple in a first attempt. Write these assumptions down and put the most useful ones as a prologue to your responses to (b) - (c).

\vspace*{.4 true cm}

Now see if we can build a model using these assumptions to help us tell how long it might take an ant to {\it extend\/} a tunnel from distance $x$ to distance $x+h$. Thus we seek an expression for (\ref{eq:change}):

\begin{equation}\label{eq:change}
T(x+h) - T(x) = \rule{4 true cm}{.01 cm}
\end{equation}

Notice that $T(x+h) -   T(x) \neq T(h)$, for $T(h)$ represents the time it takes to dig a small tunnel of length $h$ from the mouth of the tunnel, while $T(x+h) - T(x)$ includes the time it takes to extend the tunnel from length $x$ to $x+h$. This latter time  must account for the time for the ant to bring the material all they way out along a path of length $x$ from the region from $x$ to $x+h$ which is more than just the time it takes, $T(h)$, to bring the material a distance of only $h$ to the mouth of the tunnel.

\begin{itemize}
\item [b)] List the variables (present or to be introduced) on which the expression in (\ref{eq:change}) might depend.

\begin{figure}[t]
\begin{center}
\includegraphics[width=10.9cm]{1-007-ModelingOneAnt2.eps}

\vspace*{-11.5  true cm}


\parbox{12.2 true cm}{\caption{\label{fig:1-007-ModelingOneAnt2} Useful diagram for discovering the time it takes to build a small section of the ant tunnel from distance $x$ to $x+h$. }}
\end{center}
\vspace*{-.5  true cm}
\end{figure}



\item [c)] Below are several possible mathematical models for (\ref{eq:change}). Defend or reject each and offer your reasons. Perhaps modify one or two and make it better. When trying to reject a model consider some trivial cases and see if it makes sense, e.g., $h=0$ or $x = 0$  or either $h$ or $x$ very large.
\begin{itemize}
\item [i)] $T(x+h) - T(x) = x + h$.
\item [ii)] $T(x+h) - T(x) = x - h$.
\item [iii)] $T(x+h) - T(x) = x^h$.
\item [iv)] $T(x+h) - T(x) = x\cdot h$.
\item [v)] $T(x+h) - T(x) = h^x$.
\item [vi)] $T(x+h) - T(x) = c$.
\end{itemize}
\end{itemize}





\begin{itemize}
\item [d)] Convert your model difference equation~(\ref{eq:change}) to a differential equation with appropriate initial conditions.
\item [e)] Solve the differential equation you create in (d) for $T(x)$. Hint:  What initial condition $T(0)$ will you use?
\item [f)] Use your solution from (e) to determine how much longer it takes to build a tunnel which is twice as long as an original tunnel of length $L$. What would some of your original function models you set forth in (a) have told you here?
\item [g)] Suppose we had two ants digging from either side of our sand hill along the same straight line. How would this alter the total time for digging the tunnel? 
\end{itemize}

Of course, we can apply these same principles of our model to real tunnel building for engineers.

\begin{itemize}
\item [h)] If we were considering (g) as related to engineering construction of a long tunnel of length $L$, outline some of the issues we should be aware of  when having two crews  (one from each end of the tunnel) working on the tunnel.
\end{itemize}


\teacher{
\section*{NOTES FOR TEACHERS}


This is an activity which can be covered in half a class period with student engagements either as a total class or in small groups.

This is a writing assignment in which students build, defend, and solve their mathematical model and hand in as a homework assignment. It could also serve as a mini-project.  Give students time (say a week, we found is good enough time) to organize, reflect, and complete their analysis and communication in their written effort.

Students who have had calculus both differential and integral calculus could use this as an introduction to what a differential equation is, how it might come to be, and how one can solve it, with attention to initial condition use for determining the constant of integration from the calculus.

The evaluation of such student effort could be a rubric based approach designed by each teacher to fit a pre-constructed course rubric or one just for an introduction write-up which is what this would be. Material to look for are clear statements of  assumptions, connections to the necessary assumptions - both mathematical formulation and verbal narrative, connectivity between statements and sections, good communication in general,  proper use of symbols, and correct, logical, and with attention to detail mathematical exposition.  Finally, there should be some conclusion, perhaps with some reflection on the process as individual teachers might assign.

Since this is often an early assignment the teacher needs to be sure that mistakes and incomplete ideas are corrected, but also small issues and statements should be rewarded on this set of baby steps to full modeling with differential equations. Grade might reflect effort, neatness, organization, and flow as well as technical details and technical narrative. We find it is not good to give students low grades if evidence of a good faith effort to communicate is seen, especially early in such a course. This is an assignment where a ``pep talk'' with encouraging words approach with much feedback is better than a harsh grade for motivation! 

 
Here are some assumptions that come up in the consideration of modeling ant tunnel building. Some might be  simplifications, some might be oversimplifications, and some might be  irrelevant.

\begin{enumerate}
\item  The sand is of uniform nature, consisting of fine grains.
\item The sand is dry.
\item The sand is damp.
\item The ant works alone.
\item It is summer.
\item The ant works in the day time and the night time.
\item The ant will build a level straight tunnel.
\item  The ant will build in uniform, moist, fine sand.
\item  The ant will walk as fast when she is carrying something as when she is unburdened. It would take more energy to do so, but pace is the same.
\item The ant is tireless and continues to work at steady pace.
\item  The tunnel's cross-sectional area is constant.
\item The material through which the tunnel is dug is NOT sand, but rather hard clay.
\item The time to dig out the material some $x$ units into the tunnel is not insignificant.
\item The ant need not remove the sand from the opening of the tunnel.
\item  The ant is digging into the side of a sand wall.
\item Etc.
\end{enumerate}



Here is a way to get students to move along one path for a model.  Suppose we went with a variation of (iv) in our candidate list for $T(x+h) - T(x)$ above, i.e.,

\begin{equation}\label{eq:diffEq}
T(x+h) - T(x) = \alpha\cdot  x\cdot h\, .
\end{equation}


What might $\alpha$ represent?  What information from  our assumptions might be included in $\alpha$? What will the units of $\alpha$ have to be?


This activity has always proven accessible for students and with a bid of guidance the students can produce a reasonable model and test it for a reality check to their satisfaction.  The work  \cite{WinkelAnt1994} is a paper in which the modeling process is described, along with student issues.

\begin{thebibliography}{99}


\bibitem{WinkelAnt1994} Winkel, B. J. 1994. Ants, Tunnels, and Calculus:  An Exercise in Mathematical Modeling. {\it Mathematics Teacher}. 87(4): 284-287.

\end{thebibliography}



 }


\end{document}
