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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\TIME OF DEATH}




\author{Brian Winkel,
Director SIMIODE\\
Chardon OH 44023   USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
Students are asked to determine the time of death given both environmental temperature situations and two observations of body temperature under several different circumstances.
\end{abstract}



\section*{SCENARIO DESCRIPTION}

It was hot for this London fall day -- 70$^{\circ}$F. Holmes arrived at Baker Street Annex to find the
inspector hunched over the body. ``It is important that we determine the exact time of death, sir,
for in that way we may immediately determine the motive,'' said the inspector. Not wishing to
pursue the unpursuable {\it non sequitur\/}, Holmes took out his  thermometer and after a few
moments of discrete (!) investigation, announced, ``I say! 94.6$^{\circ}$F. (What, no metric system?) And
it is presently noon.'' With that he departed into the London fog, to return to the body at the
same spot in one hour. After performing another investigation Holmes declared, ``93.4$^{\circ}$F at 1
o'clock.'' And then silence\ldots

``Inspector, the murder occurred at exactly 8:58.51204 o'clock a.m. Good day to you, sir!''
Later in their chambers Watson asked, ``I say! (The British always say by saying, unlike the
Hsitirb's who say by not saying.) How did you do that, Holmes?'' ``Elementary, my dear
Watson. A simple application of a law of Newton,'' said Holmes.

``Here, here Holmes, you don't mean to say he fell out of a window?'' snapped Watson,
sensing the gravity of the issue. 

Remaining cool Holmes began at the beginning, where he always
began with Watson, ``You see \ldots ''

\subsection*{Activities}

\begin{enumerate}
  \item Use a mathematical model to either refute or corroborate Sherlock Holmes' conclusion. Be sure you state your assumptions and defend the use of your model.
  \item Suppose the forensic folks confirm that the victim had viral evidence of fever with a body temperature of  103.2$^{\circ}$F at the time of death. How does this change the evidence you offer in (1)?
  \item What assumption are you  making about the environmental temperature during the period from death until discovery? Drop this simplifying assumption and offer up a more realistic model. Then address (1) again in this context. How does this change the evidence you offer in (1)? How does Holmes predictive abilities look now? Hint:  See Table~\ref{table:LondonTemp}.
   
 \begin{table}[h]
\begin{center}
\small\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \hline
 Time &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 &12 \\ \hline
 Temp & 60.8 & 59. & 57.2 & 55.4 & 55.4 & 53.6 & 55.4 & 57.2 & 59. & 62.6 & 66.2 & 68. &   69.8 \\ \hline \hline
 Time &  13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 &\\ \hline
 Temp &71.6 & 73.4 & 73.4 & 73.4 & 69.8 & 69.8 & 68. & 66.2 & 62.6 & 59. & 57.2 & 55.4 &\\ \hline \hline 
\end{tabular}
\parbox{9 true cm}{\caption{\label{table:LondonTemp} London temperature in degrees Fahrenheit at hours past midnight  ($t = 0$) on the day of the murder.}}
\end{center} 
\end{table}
 \end{enumerate}
  

\teacher{\section*{NOTES FOR TEACHERS}


This material describes  a  use of Newton's Law of Cooling to determine the time of death. Much has been written on this subject \cite{Leinbach2011, Knight1995}. While the data is not from an actual case it is reasonable and realistic. We have used this material which we wrote up in 1979 \cite{Winkel1979} for years. Students enjoy seeing Newton's Law of Cooling in action and come to understand its reasonableness, especially when we attribute a higher body temperature at the time of death due to fever and set the scene into a more realistic temperature regime which is changing.

When we have done material surrounding Newton's Law of Cooling we find students can invent the Law and we do not have to give it to them. It is a reasonable consequence of thinking how things cool, e.g., their coffee in a hot room vs. in a cold room. They then own the model and we do not tell them it is Newton's Law of Cooling, rather that it is {\it TheirName\/}'s Law of Cooling!  The narrative, both mathematical and descriptive, in the Mathematica notebook 1-45-T-TimeOfDeath-TeacherVersion.nb and the accompanying pdf version for non-Mathematics users contains details and ideas which teachers can use in assisting students develop their model and implement their analyses.

In Activities (1) and (2) students often catch on to the fact that the temperature IS NOT constant through the morning. However, we ask them to proceed as if it were and point them to Activity (3) which reflects their understanding of reality. We commend them for offering quite good assumptions and checking the ``conventional'' constant temperature assumption.
 
In Activity (3) we attempt to offer students the opportunity to model the changing temperature in order to more realistically model the environmental temperature, $TE(t)$, as non-constant using Newton's Law of cooling, $ T'(t) =  k (TE(t) - T(t))$, for the temperature of the body, $T(t)$ with rate constant $k$. Students need to have reasoned estimates \cite{Winkel2012} of the parameters $a$, $b$, $c$, and $d$ in order to attain a good fit in any system, be it Mathematica, Maple, SAGE, or EXCEL.

Finally, depending upon the working assumptions, students find Holmes ``spot on'' as the English say or way off!

\begin{thebibliography}{99}

\bibitem{Knight1995} Knight, B. (ed.) 1995.  {\it The Estimation of the Time Since Death in the Early Postmortem Period\/}. 
London:  Edward Arnold.

\bibitem{Leinbach2011} Leinbach, C. 2011.  Beyond Newton's law of cooling -
estimation of time since death. {\it International Journal of Mathematical
Education in Science and Technology\/}. 42(6): 765-774.

\bibitem{Winkel1979} Winkel, B. 1979. Elementary (My Dear Watson) Differential Equations. {\it Mathematica Magazine\/}. 52(5): 315. 

\bibitem{Winkel2012} Winkel, B. J. 2012. Shedding a Little (Sun)Light on Data Analysis and Technology. {\it International Journal of Mathematical Education in Science and Technology\/}. 43(3): 375-381. 


\end{thebibliography}
}








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