% SIMIODETemplate-TeacherORStudent.tex and simiode.cls for LaTeX2e
% Version 29 July 2015.
%
% This class (simiode.cls) is a modification of that used by
% the journal PRIMUS - Problems, Resources, and Issues in
% Mathematics Undergraduate Studies.
%
% See http://www.tandfonline.com/loi/upri .
%
% The class was developed by  Marc Frantz with suggestions of the
% PRIMUS Editorial Board, especially John Thoo.
%
% This template is to be used with the  class file "simiode.cls".
%
% The latest version of this class and template files
% can be found at
%
%       <http://www.SIMIODE.org> in the Author section.
%
% A "Quick Start Guide" appears below.
%
% This file must be processed with version LaTeX2e or higher.
%
\documentclass{simiode}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
%
% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
%
%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
%
% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review (but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
%
%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.
% Rename the pdf after you compile each final version.
% This option was created  by Eric Sullivan, Carroll College, Helena MT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding double spacing %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manuscripts should be submitted doubled spaced.  This is
% accomplished via the \baselinestretch command below.
% However, you may obtain an idea of how your article will
% look in final form, so that you can better plan your
% presentation.  To see your manuscript formatted very
% similarly to the on-line published format of  SIMIODE,
% comment the following line out by inserting a percent sign % (%) at its beginning.

\renewcommand{\baselinestretch}{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in
% the top matter. Likewise, any environment definitions such
% as \newtheorem or \newenvironment should also go in the
% top matter.
%
% Do not \input your macros as separate files within your
% final version because more files make it harder for the
% editors and users to keep track of and/or modify and
% append your paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\M\&M -  DEATH  AND IMMIGRATION MYSTERY}




\markboth{M\&M - Death  and Immigration Mystery}{M\&M - Death  and Immigration Mystery}






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


\makeStitlePDFLaTex  


\begin{abstract}
We describe a classroom activity in which students use M\&M candies to simulate death and immigration. Each student conducts an experiment with an immigration rate unique to that student - of that student's choice. Collected data on generation or iteration and population  is then passed to another student and the receiving student builds a mathematical model, usually a linear first order, difference or differential equation, and estimates the death and immigration parameters.
\end{abstract}


\section*{SCENARIO DESCRIPTION}


Consider a regular size bag of M\&M candies, not peanut, just regular. Usually there are about 55 pieces in each bag. They are of different colors, but each piece almost always has an 
``m" pressed on one side and not the other. Hence, there are distinctions between the sides. Let us conduct an experiment (a  death and immigration simulation) on the M\&M's. 

\subsection*{Death and Immigration Mystery Model}

You will conduct this simulation in which M\&M's either live or die, while at each iteration a certain number (between 6 and 16) of M\&M's (your choice of number, call that number $MI$) immigrate into the population. After opening the bag of candy  your teacher will tell you to place an original number, $M(0)$, of M\&M's pieces into a small cup, call this cup the Active cup, to start the simulation. You should have a second Reserve cup for reserve M\&M's and a paper plate for capturing the M\&M's as they are tossed. Use the sheets provided  for recording your data - one copy for you and one copy to pass along to another student.

  

\begin{quote}
\vspace*{.2 true cm} {\bf The Experiment:} 
Gently shake the M\&Ms out onto the paper plate to catch the M\&Ms. We determine for each M\&M if it lives or dies.
If the ``m'' shows on top this M\&M goes ``belly up'' and dies, otherwise there is life for this M\&M. Upon death you should remove the dead M\&M's from the population (set these aside into the Reserve cup as we may need them). THEN you select a number of M\&M's from the Reserve cup to immigrate back into the population, call that number $MI$. {\bf Use this same number of immigrants in each generation  you conduct}. Your immigration number should be between 6 and 16.  Add these immigrants to the surviving M\&M's in the plate.  Now count and note down the number of M\&M's who survive or have immigrated as the number of M\&M's at the start of the next generation in Table~\ref{table:BD} or on sheets provided to you. Then put these M\&Ms back into your Active cup for the next iteration. Do this over and over for, say, 10 generations or iterations. 
\end{quote}




\begin{itemize}
\item [a)] Before starting the experiment describe what you think will happen. Care to make a prediction?
\item [b)] State your assumptions about the physical activity.  Offer up some assumptions in support of your description and some which might not have that much to do with what you expect to happen.   Many times  assumptions are run into each other, so break the assumptions down to their simplest form, i.e.  do not use  words like ``and." 
\item [c)] In Table~\ref{table:BD} record what happened and compare this with what you thought would happen.  You may also record the iteration or generation and \# M\&Ms at start of iteration in two columns on both index cards. Label the columns at the top (Generation and \# M\&M's) as shown in  Figure~\ref{fig:1-1B-FrontCard}  of one such simulation. On the back of the index card, in tiny print, in the lower right, in very small print (we are stressing small print), put the number of immigrants YOU used in each iteration. This number is the mystery to be discovered with modeling by the student to whom you pass along one of your index cards. It will help  confirm the  model with the data you pass along.

\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|} \hline\hline
\multicolumn{2}{|c|}{Modeling Death of M\&Ms} \\ \hline\hline
Iteration & \# M\&Ms at start of iteration\\ \hline\hline
0 & \\ \hline
1 & \\ \hline
2 & \\ \hline
3 & \\ \hline
4 & \\ \hline
5 & \\ \hline
6 & \\ \hline
7 & \\ \hline
8 & \\ \hline
9 & \\ \hline
10 & \\ \hline
\end{tabular}
\caption{\label{table:BD} Modeling death and immigration of M\&Ms.}
\end{center}

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


\item [d)] Compare your description/prediction with what actually happened.

\item [e)] Indicate which of your assumptions were reasonable and played a role in the experiment.
\end{itemize}


\subsection*{Model Building and Parameter Estimation}

Let us attempt to build a mathematical model of this activity, being sure to include death and immigration aspects. Suppose we were to define the number of M\&Ms alive at the start of iteration $n$ as $a(n)$.  What values would $n$ take?  What would $a(0)$ be in this case? Do you ``know" what $a(1)$ would be?  

Based on the observations and your assumptions, produce a reasonable formula for $a(n)$, i.e. offer up a discrete function $a(n)$, in the one variable, $n$, for $n = 0, 1, 2, \ldots$ . Incidentally such a model for $a(n)$ could  be in {\it closed form}, something like $a(n) = n^2 - n$, or it could be in the form of a {\it recursion} equation $a(n) = 3 a(n-1) -7$. Which form is easier to construct for this simulation?
\begin{itemize}
\item [f)] Discuss with colleagues around you the reasonableness of your function models for $a(n)$.
\item [g)] How will you measure your ``success'' as a modeler in this situation? Check with others in the class and see if you are on target with your model.
\item [h)] Use your model to the estimate immigration rate.  You might either presume death rate (what value? and in what manner?) or attempt to estimate that as well.
\item [i)] Go ahead and check on just how good your model is at predicting your experiment and determining the mystery immigration rate. Also,  re-examine your rationale in (f).
\end{itemize}


\section*{APPENDIX-SAMPLE SIMULATION DATA}

\begin{figure}[!h]
\centering
\includegraphics[width=4cm]{1-1B-FrontCard.eps}

\parbox{8.2 true cm}{\caption{\label{fig:1-1B-FrontCard} Sample generation and number of M\&M's in each generation from death and immigration simulation.}}
\end{figure}


\teacher{
\section*{NOTES FOR TEACHERS}


 

Ask the students  to describe what happens in their simulation. Soon they see that the  population model with death AND immigration never really gets to zero. Indeed, just as it appears that the population might be going to zero, our rule says, ``add a fixed number of M{\&}M's  (ALWAYS the same fixed number at each iteration)." Thus, we would not say the population goes to 0, and from the data, neither would we say the population goes to the fixed number of immigrants added each generation. It is a good idea to interrupt the class and let them summarize their progress.


This is a very good activity to permit the introduction of the notion of {\it Equilibrium Value\/}, for suppose the students immigrate 10 each iteration. Then their data seems to get to 20 and roughly stay at or near 20. Is there some value, say EV, such that  when we are at EV in the next iteration or generation we find ourselves at EV again in this model, i.e.~can we find EV such that ${\rm EV} = \frac{1}{2}{\rm EV} + 10$?  This is algebra and we solve for EV to obtain ${\rm EV} = 20$ -- the very value their simulations seem to be dancing about.


Students can develop the immigration model $a_{n+1} =  \frac{1}{2}a_n + 10$ with $a_0 = 50$ 
(although we  started our simulation with $a(0) = 6$ in Figure ~\ref{fig:1-1B-BackCard}) for $n = 0, 1, 2, 3, \ldots$ and iterate it by hand (or in Excel) to compare the model's prediction to the data. A similar approach is to develop the difference equation model $\Delta a_n=a_{n+1} - a_n = -\frac{1}{2}a_n + 10$ or if the students are ready for it, the differential equation model, $a'(t) = -\frac{1}{2} a(t) + 10$, both with attendant solution strategies and initial conditions of $a_0 = 50$ or  $a(0) = 50$, respectively.

If one is interested in teaching the closed form solutions for either the difference or differential equation approach the students can compare the analytic results with the data. The closed form solution of the difference equation is $$a_n = 20\left(1 +   2^{2-n}\right)$$ while the comparable solution for the differential equation is $$a(t) = 20 \left(1- e^{-t/2}\right)\, .$$ Of course, the analytic solution strategy could precede the modeling activity to reinforce the use of the analytic model. That is the call of the teacher.


In  \cite{Winkel2014} we offer an opportunity for students to collect data from a simulation we conducted and for which we offer generational slides with the dying M\&M's removed and immigrating M\&M's put in place at each generation. Here students can collect the data from the screen if a physical simulation with real (yummy) M\&M's is not possible or feasible. In this simulation  the immigration rate of 10 is given, but students will need to develop a scheme for recovering this number without knowing it.

\subsection*{Conclusion}

Through physical simulation of a ``real" population students collect data for a mathematical model of a physical  phenomena and then compare the results of the  model to the data. They also get to examine data from a fellow student and attempt to determine the mystery immigration rate offered by one fellow student though data. The opportunity is rich with many parallels to modeling in the real world and the students can see at each step the assumptions, data collection, modeling, and comparisons. We recommend the reader try this activity and perhaps add something to it.

We provide a complete data sheet with instructions below and we recommend you print two copies for each student and give them to the students with their ``supplies."


In Figure~\ref{fig:1-1B-BackCard} we display the back of the card for the data offered in Figure~\ref{fig:1-1B-FrontCard} in which the immigration level is specified. This is the ``mystery'' immigration number the students seek if they use Figure~\ref{fig:1-1B-FrontCard} data and not their own generated data. 

\begin{figure}[!h]
\centering
\includegraphics[width=5cm]{1-1B-BackCard.eps}
\parbox{11.8 true cm}{\caption{\label{fig:1-1B-BackCard} Back of data card with number of immigrants per each generation in lower right of card for sample generation and number of M\&M's in each generation from death and immigration simulation in Student Version of this modeling scenario.}}
\vspace*{-.4 true cm}
\end{figure}


\vspace*{-.4 true cm}
\subsection*{Acknowlegements}
The essence and detail of this approach appears in complete detail in \cite{WinkelMM2010}. We acknowledge and appreciate the feedback from many, many colleagues who have tried this approach and activity in their teaching and have made the discussion a richer one.  The project NEVER FAILS to engage, teach, and interest students, and it often serves as a full-scale first introduction to modeling and technology in the study of  differential equations.

In particular, we thank Dina Yagodich, Frederick Community College, for suggesting that we start with a few M\&M's, say 6, instead of our original number of 50. This cuts down on the time for the simulation, reduces the tedium of counting so many pieces, and arrives ``at or near" an equilibrium in short order. 
This modeling activity should take less than 30 minutes in class. 
\vspace*{-.2 true cm}

\begin{thebibliography}{1}

\bibitem{Winkel2014} Winkel, B. J. 2014. 1-1-M\&M-Death-ImmigrationSimulationImageData. .

\bibitem{WinkelMM2010} Winkel, B. J. 2010. Population Modeling with M\&Ms.  {\it International Journal of Mathematical Education in Science and Technology\/}. 40(4): 554-558. \href{http://www.tandfonline.com/doi/abs/10.1080/00207390802641668\#.V3FucqIxC9Y}{http://www.tandfonline.com/doi/abs/10.1080/00207390802641668\#.V3FucqIxC9Y}. 
\end{thebibliography}
}

 \newpage
\begin{center}
{\bf Death and Immigration Mystery Simulation}
\end{center}

\subsection*{Material}

\begin{center}
\begin{tabular}{ll}
1 small bag M\&M's (regular not nut) & 
2 small cups (paper cups will do)\\ 
1 paper plate & 
2 copies of this sheet for tallying results\\
\end{tabular}
\end{center}
\subsection*{Simulation instructions}

You are going to conduct a simulation with M\&M's as members of your  population. The population will experience death and immigration for a number of generations, say 10 as outlined in the table below,  and you will record the population at each generation  in table below.

Your teacher will assign you an original number of M\&M's, $M(0)$, and you will  designate a number of immigrants (number between 6 and 16) to use in each generation, $MI$, Write that number, $MI$, in small font at lower right hand corner  on the back of this sheet.

You will conduct your simulation and record the same data on both copies of this sheet, one for your records and one to pass along to a colleague for analysis.

Steps of simulation:
\begin{enumerate}
\item Open a bag of M\&M's and place $M(0)$ (instructor designated initial population) M\&M's in one cup (Active cup) and the remaining M\&M's in the other cup (Reserve cup).
\item Write in $M(0)$ in table below.
\item Gently toss  the M\&M's from the Active cup onto the paper plate and remove the M\&M's which have an ``m" facing up, placing them  into the Reserve cup.
\item Add $MI$ immigrants to the plate from Reserve cup and count the  M\&M's on the plate.
\item Enter that tally in the next generation's column designated \# M\&M's at start of iteration.
\item Place all M\&M's from the plate into the Active cup and go to Step (3) above. 
\end{enumerate}
  
\vspace*{.1 true cm}


\begin{center}
\begin{tabular}{|p{4 cm}|p{6 cm}|} \hline
\multicolumn{2}{|c|}{Modeling Death of M\&Ms} \\ \hline\hline
Generation/Iteration & \# M\&Ms at start of iteration\\ \hline\hline
\multicolumn{1}{|c|}{0} & $M(0) = $ \\ \hline
\multicolumn{1}{|c|}{1} & \\ \hline
\multicolumn{1}{|c|}{2} & \\ \hline
\multicolumn{1}{|c|}{3} & \\ \hline
\multicolumn{1}{|c|}{4} & \\ \hline
\multicolumn{1}{|c|}{5} & \\ \hline
\multicolumn{1}{|c|}{6} & \\ \hline
\multicolumn{1}{|c|}{7} & \\ \hline
\multicolumn{1}{|c|}{8} & \\ \hline
\multicolumn{1}{|c|}{9} & \\ \hline
\multicolumn{1}{|c|}{10} & \\ \hline
\end{tabular}

\end{center}







\end{document}
