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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION\else STUDENT VERSION \fi\\ PROPAGATION OF THE WORD JUMBO }



\markboth{Word Propagation}{Word Propagation}






\author{
\begin{minipage}[t]{6 true cm}
Rachel L.~Bayless\\
Agnes Scott College\\
Decatur GA  30030USA\\
rbayless@agnesscott.edu
\end{minipage}
\  \quad \
\begin{minipage}[t]{6 true cm}
Rachelle C. DeCoste\\
Wheaton College\\
Norton MA02766 USA\\
decoste\_rachelle@wheatoncollege.edu
\end{minipage}
}

\makeStitlePDFLaTex






\begin{abstract}
This activity is a gentle introduction to modeling via differential equations. The students will learn about exponential growth by modeling the rate at which the word jumbo has propagated through English language texts over time. 
\end{abstract}



\section*{SCENARIO DESCRIPTION}

In 1861, the elephant who became known as Jumbo was captured in Ethiopia and purchased by animal collector Johan Schmidt.  First known as the ``Children's Giant Pet,'' the largest elephant known at the time was renamed Jumbo, a word believed to have been created from the combination of the Swahili words {\it jumbe} and {\it jambo}, meaning ``chief" and ``hello", respectively. \cite{barnum}

Eventually Jumbo was traded to the London Royal Zoological Society.  By age seven the 11.6 foot, 6.5 ton bull's daily intake included ``200 pounds of hay, 1 barrel of potatoes, 2 bushels of oats, a slew of onions and several pails of water.''  To address health concerns, his zookeeper Matthew Scott gave Jumbo a gallon of whiskey per day when he felt it was necessary. \cite{barnum}


Despite much protest from English residents, in 1882 Jumbo was acquired by P.T. Barnum to be added to his circus, ``The Greatest Show on Earth.''  For three years, until his death in 1885, Jumbo traveled with the circus on a rail car that was specifically made to be large enough to hold the elephant.  After his death, Jumbo's hide was stuffed and continued to tour with the circus until it was donated to Tufts University for display in the Barnum Museum of Natural History.  Tufts' athletic teams were named after the elephant, and his hide was seen as a source of good luck.   Jumbo's bones were mounted and displayed in the American Museum of Natural History in New York. \cite{tufts}

Today the word jumbo is commonly understood to mean very large, as in jumbo jet and jumbo shrimp.  Through this project we will explore how the word jumbo has spread through the English language based on its original use in the 19th century.

In your Calculus II class, you may have been briefly introduced to differential equations; some differential equations can be solved with integration.  One common model studied is population growth.  The most basic model assumes that  population growth should be proportional to the size of a population.  This leads to the following differential equation:  $$\frac{dP}{dt}=kP$$ where $P(t)$ is the population at time  $t$ and $k$ is the constant that is determined by how quickly the population grows.  You may want to keep this example in mind as you consider the following questions with your group.


\begin{enumerate}
\setlength\itemsep{1.5em}
\item How do you quantify a word's popularity? 

\vspace{\fill}



\item Suppose you have a sentence consisting of 10 words, and the word jumbo appears exactly once. On the other hand, suppose you have a paragraph consisting of 100 words, and the word jumbo appears exactly once. Does this example change your definition of popularity? 

\vspace{\fill}

\item We want popularity to be a function of time.  Let us define $t$ to be years.  The word jumbo first appeared in print in 1861; thus our function will only be defined for $t\geq 1861$.  Then define popularity as a function of time $t$  by  $P(t) = $ 
{  \it Percentage of words published in books in year $t$ that is the word jumbo}. Explain why, specifically, $$P(t)=\dfrac{{\rm \# \ of \ instances \ of \ jumbo \ in  \ books \ published \ in \ year \ } t}{{\rm \# \ of \ words \ published \ in \ books \ in \ year \ } t}\times 100$$ provides a reasonable measure of popularity. 


\vspace{\fill}

\item Explain (in words) what $P(1900) = 2.08 \times 10^{-6}$ means. 

\vspace{\fill}

\item Explain (in words) what $\dfrac{dP}{dt}$ represents. 

\vspace{\fill}

\item Explain (in words) what factors or variables could influence $\dfrac{dP}{dt}$ and how. 

\vspace{\fill}



\item Consider the scatter plot (Figure~\ref{fig:1-32-JumboGraphNoCurve}) of data from Google Ngrams \cite{google} along with your answer to the question above. Given this new information, what factor appears to be influencing $\dfrac{dP}{dt}$? 

\begin{figure}
\begin{center}
\includegraphics[width = 11 true cm]{1-32-JumboGraphNoCurve.eps}

\vspace*{-9.7 true cm}
\parbox{9 true cm}{\caption{A scatter plot of percentage of printed words that are the word jumbo versus time for the years 1880 to 1940. }}
\end{center}
\label{fig:1-32-JumboGraphNoCurve}
\end{figure}

 


\item Write a mathematical expression for $\dfrac{dP}{dt}$.

\vspace*{1 true cm}


\noindent Consider the following differential equation, 

\begin{equation}\label{DE}
\frac{dP}{dt} = kP,
\end{equation}
where $P(t)$  is the percentage of all words published in books in year $t$ that is the word jumbo.

\item Explain (in words) what it means if $k = .05$.

\vspace{\fill}

\item Explain (in words) what it means if $k < 0$.

\vspace{\fill}


\item Solve the differential equation in (\ref{DE}). That is, find a function $P(t)$ such that its derivative is a constant multiple of the function itself. (There is more than one way to do this. Explain your thinking.)

\vspace{\fill}

\item Returning to thinking of $P(t)$ as the percentage of total words in books that is the word jumbo, draw the general shape of the graph.

\vspace{\fill}

 
\item What does the model forecast will happen as $t$ increases? Does the model continue to be valid? Explain.




\vspace{\fill}
\newpage

\end{enumerate}

We would now like to check that our theoretical model fits the real data from Google Ngrams. First, Table~\ref{table:JumboDataNumbers} is an excerpt from the raw data \cite{google}. It shows the year, total words published, the number of times jumbo appeared, and the percentage of the published words that are the word jumbo. 



\begin{table}
\centering
\begin{tabular}{ |c|c|c|c| } 
\hline
 Year & Total Words & Jumbo Appearances & Percentage  \\ 
\hline
\hline
 1880 & 784223075 & 6 & $7.65088428442379 \times 10^{-9}$ \\ 
\hline
 1881 & 789254798 & 12 & $1.52042154579338 \times 10^{-8}$ \\ 
\hline
 1882 & 828502461 & 1 & 1$1.20699701820198 \times 10^{-9}$ \\ 
\hline
 1883 & 930196929 & 17 & $1.82756999835247 \times 10^{-8}$ \\ 
\hline
 1884 & 881638914 & 3 & $3.40275361302847 \times 10^{-9}$ \\ 
\hline
\end{tabular}
\parbox{8.6 true cm}{\caption{\label{table:JumboDataNumbers}The first five rows of a spreadsheet with the data on yearly usage of the word jumbo in published books. }}
\end{table}



From these data we can create a scatter plot of percentage of printed words that are the word jumbo versus time which is overlaid with an appropriate exponential. 



\begin{figure}
\begin{center}
\includegraphics[width=11 true cm]{1-32-JumboGraph.eps}

\vspace*{-9.7 true cm}
\parbox{9 true cm}{\caption{A scatter plot of percentage of printed words that are the word jumbo versus time for the years 1880 to 1940. The red line is the best-fit exponential, $P(t) = 5.07231223e^{.0549975t}$.}}
\label{1-32-JumboGraph}
\end{center}

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



 


\teacher{
\section*{NOTES FOR TEACHERS}

This project is an in-class guided discussion that is appropriate for the first day of an introductory differential equations class.  Students should be placed in groups of 3-4 to work on the questions.  Students should be given a few minutes to work on each question with their group, then the class should reconvene for whole class discussion to ensure students are on the right track with each concept before moving on to successive questions. Discussion following each question is necessary since this is a first day activity, and the authors assume little prior knowledge about differential equations.  Depending on the instructor's classroom management and the depth with which the class discusses each question, this activity could take 20-50 minutes. 

The main assumption in this modeling activity is that the rate at which a word \emph{initially} propagates through the verbiage is proportional to how often the word is being used, so the major differential equations concept covered is exponential growth.  It may not be obvious that students would immediately assume exponential growth.  However, since this is a first day activity, our assumption is that students will have seen very few differential equations prior to this class.  They will likely have been exposed to exponential growth, and we lead them in this direction by reminding them of a common example from Calculus II, population growth.  This activity places an emphasis on introducing students to many other universally important aspects of modeling, and we do not assume that the students will have had much experience at transcribing an open-ended scenario into the language of mathematics. During completion of this activity the students will:

\begin{enumerate}
\item Explain the assumptions that lead to an exponential model. \label{exp_model}
\item Carefully define and use a dependent variable that is a percentage rather than a raw count.
\item Understand the form of a solution to a differential equation and check that a given function is a solution.
\item Solve a differential equation using separation of variables. 
\item Analyze the efficacy and limitations of a differential equation model within the context of a real-world scenario.
\item Discuss limitations of modeling and choice of model, making the point that there is not necessarily a ``right'' answer, but that some models will work better than others, depending on the data.  
%In particular, a common limitation is that a model may work will for a short time, but does not remain valid for all time. 
% Additionally, students should recognize that modeling may require piece-wise functions, though the details of this should be left to a future lesson.  
%Discuss the fact that there is not necessarily a ``right'' or ``wrong'' answer in modeling. 
\item Recognize a connection between differential equations and linguistics (two seemingly unrelated fields).
\end{enumerate}

\noindent  \textbf{Student Questions:} The following questions exactly match those in the Statement section, but here we have interspersed italicized teacher comments. These questions can be asked of the students either by passing out the student version of this worksheet or using them as conversation starters in a discussion-style class. In the teacher comments we provide information on the motivation for asking each question, suggestions for classroom management, and the solutions.  

\begin{enumerate}
\item How do you quantify a word's popularity? 

\vskip.1in {\it  Let the students work in small groups to discuss this for about 5 minutes. Students will likely come up with an idea of counting word appearances somewhere (i.e., Google searches, social media posts, etc.). Have a group discussion about how counting the number of times a word is used in literature is one way to measure ``usage''  and ``popularity'' in society.  This is one reasonable suggestion because of the availability of data, specifically through the Google Ngram viewer \cite{google}.  At this site, the user can find data on how many times any word or phrase is found in a sample of all English language books for a span of user-selected years. The instructor should talk about this site, but may  not want to show the students the site yet because the initial display of data is through a graph (to download raw numbers, click on  ``About NGram Viewer'' at the bottom of the page).  Showing the graph of any word propagation may give students answers or hints to the remaining questions concerning the dependent variable and the shape of the curve.}  




\item Suppose you have a sentence consisting of 10 words, and the word jumbo appears exactly once. On the other hand, suppose you have a paragraph consisting of 100 words, and the word jumbo appears exactly once. Does this example change your definition of popularity? 

\vskip.1in  {\it  The students are likely to be thinking about counting the raw number of times a word appears based on question 1. The goal at this point is to get them thinking about percentage of all words printed. Percentage is favorable to raw counts because raw counts may have the confounding variable of more books being printed over time.  The raw count of most words should therefore increase after the initial introduction of a word, but this does not necessarily represent overall popularity as we want students to think about it.  }

\item % Carefully define the variable you want to use to represent the propagation of the word jumbo. 
We want popularity to be a function of time.  Let us define $t$ to be years.  The word jumbo first appeared in print in 1861; thus our function will only be defined for $t\geq 1861$.  Then define popularity as a function of time $t$  by  $P(t) = $ 
{\rm Percentage of words published in books in year} $t$ {\rm that is the word jumbo}. Explain why, specifically, $$P(t)=\dfrac{{\rm \# \ of \ instances \ of \ jumbo \ in  \ books \ published \ in \ year \ } t}{{\rm \# \ of \ words \ published \ in \ books \ in \ year \ } t}\times 100$$ 
 provides a reasonable measure of popularity.


\vskip.1in {\it One of the main goals of this lesson is careful definition of variables, so questions of this nature will be repeated throughout. It is a new (and often confusing) concept for students to model the percentage of a population as the dependent variable (as opposed to a total population).}


\item Explain (in words) what $P(1900) = 2.08 \times 10^{-6}$ means. 

\vskip.1in {\it If $P(1900) = 2.08 \times 10^{-6}$, then $2.08 \times 10^{-6}$ is the percent of all words printed in the year 1900 were the word jumbo. Again, part of the goal of this lesson is careful definition of variables, so this question is intended to use a concrete example to reiterate the definition of $P(t)$ as a percentage.}

\item Explain (in words) what $\dfrac{dP}{dt}$ represents. 
\vskip.1in {\it This is the instantaneous rate of change of the percentage of words published in year $t$ that is the word jumbo.  If $\dfrac{dP}{dt}>0$ for a particular value of $t$, then the percentage of all words that is jumbo is increasing at year $t$. That is, jumbo is being used more often when compared to all words printed in the literature for that year. }

\item Explain (in words) what factors or variables could influence $\dfrac{dP}{dt}$ and how. 
\vskip.1in {\it Here students are encouraged to come up with many different explanations. Some may think about societal movements, such as political movements, social movements, or even turmoil/war. They may also think about how certain words come into fashion for a short period, then their popularity dies off. Additionally, they may think about the classic calculus problem involving rumor spread, and that the more people who hear a rumor affects the amount that it is told.}

\item Consider the following scatter plot (Figure~\ref{fig:1-32-JumboGraphNoCurvea}) of data from Google Ngrams \cite{google} along with your answer to the question above. Given this new information, what factor appears to be influencing $\dfrac{dP}{dt}$? 


\begin{figure}[!h]
\begin{center}
\includegraphics[width = 11 true cm]{1-32-JumboGraphNoCurvea.eps}
\vspace*{-10.7 true cm}

\end{center}
\label{fig:1-32-JumboGraphNoCurvea} 
\end{figure}

\begin{center}
\parbox{9.2 true cm}{{\bf Figure 3.} A scatter plot of percentage of printed words that are the word jumbo versus time for the years 1880 to 1940. }
\end{center}



\vskip.1in {\it At this point, the students will see that exponential growth may be an appropriate model of word propagation. We can argue that the presence of a word in the literature could create more instances of the word in future books.  This means that the rate of change of $P(t)$ may be proportional to $P(t)$.  It is likely that students will have seen this concept before, but if not, then they can still reason through the main idea. Students may need a reminder that ``proportional to'' means ``a multiple of'' which will lead to the answer of the following question.}


\item Write a mathematical expression for $\dfrac{dP}{dt}$.
\vskip.1in {\it  Here it is particularly important to make sure that the whole class has discussed the previous questions together.  Without introduction to the data, it may not be obvious that the rate of change should be proportional to $P$.  In fact, students may have had other suggestions. But, it is important to consider all suggestions in light of the graph above, and in the end decide to start with a ``basic'' model and later consider the quality of that model.  So we suggest that at this point the class should agree on   $\dfrac{dP}{dt}=kP(t)$ for some constant $k$.\\}
\vskip.1in
 \noindent\rule{14.5cm}{0.4pt}\\ {\it  In the student handout we have inserted a forced page break at this point because the next page contains information that would prematurely give students answers to previous questions.  The students would initially only receive the questions through number 7.  Once the class has discussed questions 1-7, students would receive a handout with the following questions. 
\noindent\rule{14.5cm}{0.4pt}}

\noindent Consider the following differential equation, 

\begin{equation}\label{DE1}
\frac{dP}{dt} = kP,
\end{equation}
where $P(t)$  is the percentage of all words published in books in year $t$ that is the word jumbo.

\item Explain (in words) what it means if $k = .05$.

\vskip.1in \teacher{{\it This is a point where it might be particularly useful if the students first think of $P(t)$ as a total population, then change their thinking back to the context of this problem. For instance, suppose that $\dfrac{dp}{dt}=.05p$ where $p(t)$ is the number of fish in a pond at year $t$.  The differential equation tells us that the population of fish is increasing by 5\% per year at year $t$. }}


\item Explain (in words) what it means if $k<0$.

\vskip.1in {\it If $k<0$, then the percentage of printed words that is the word jumbo is decreasing from year to year.}



\item Solve the differential equation in (\ref{DE1}). That is, find a function $P(t)$ such that its derivative is a constant multiple of the function itself. (There is more than one way to do this. Explain your thinking.)

\vskip.1in {\it Let the students work on this in their groups for about 5 minutes. It is likely that some students will immediately recognize the solution as an exponential function due to their calculus knowledge.  Other students may have seen some differential equations in a calculus class and therefore may attempt to separate the variables and integrate. We suggest having a class discussion about each groups' solution method. Students' proposed solutions may or may not be correct, and this will naturally lead to  the definition of solution of a differential equation and a discussion about verifying solutions to a given differential equation. Depending on your specific course goals/schedule this would be a good point to formally introduce separation of variables. The family of solutions to equation (\ref{DE1}) is $P(t) = Ce^{kt}$, where $C$ is a constant.  }

\item Returning to thinking of $P(t)$ as the percentage of total words in books that is the word jumbo, draw the general shape of the graph.

\vskip.1in {\it Here, based on the solution discussion above,  students should draw an exponential curve with positive $k$.  This is a good time to discuss/reaffirm the definition of the dependent variable. Students should be reminded of the background of the word jumbo as a prompt about when the time line should start and probably arrive at 1861 as a reasonable starting point.  A discussion about reasonable values for the dependent variable is also appropriate here.  The students should consider how large $P(t)$ should reasonably be. }

\item What does the model forecast will happen as $t$ increases? Does the model continue to be valid? Explain. 

\vskip.1in {\it $P(t)$, as defined, is a percentage, so it cannot exceed 100. Additionally, if there exists a $t^{*}$ such that $P(t^{*})=100$, then at year $t^{*}$ 100\% of the published words are the word jumbo. Clearly this is an absurd idea.   Point out that this does not necessarily mean that the model is ``wrong'' or ``bad'' but that there are limitations to the model. In particular, an exponential model may accurately reflect the initial period of growth (much like a standard population growth model), but after some finite amount of time the model would need modifications.  This leads to the necessity of a piece-wise function, which can be briefly discussed, but is not the point of the first day discussion. }

\end{enumerate}


\noindent\rule{14.5cm}{0.4pt}\\ In the student handout we have inserted another forced page break at this point because the next page contains information that would prematurely give students answers to previous questions. If instructors have access to a computer with a projector, then the next page may be displayed (rather than printed as a handout) to save paper.  

\noindent\rule{14.5cm}{0.4pt}

We would now like to check that our theoretical model fits the real data from Google Ngrams. First, Table~\ref{table:JumboData} is an excerpt from the raw data \cite{google}. It shows the year, total words published, the number of times jumbo appeared, and the percentage of the published words that are the word jumbo. 

\vskip.2in

\begin{table}
\centering
\begin{tabular}{ |c|c|c|c| } 
\hline
 Year & Total Words & Jumbo Appearances & Percentage  \\ 
\hline
\hline
 1880 & 784223075 & 6 & $7.65088428442379 \times 10^{-7}$ \\ 
\hline
 1881 & 789254798 & 12 & $1.52042154579338 \times 10^{-6}$ \\ 
\hline
 1882 & 828502461 & 1 & 1$1.20699701820198 \times 10^{-7}$ \\ 
\hline
 1883 & 930196929 & 17 & $1.82756999835247 \times 10^{-6}$ \\ 
\hline
 1884 & 881638914 & 3 & $3.40275361302847 \times 10^{-7}$ \\ 
\hline
\end{tabular}
\parbox{12.3 true cm}{\caption{The first five rows of a spreadsheet with the data on yearly usage of the word jumbo in published books. {\it Teacher Comment: Please see the supplemental Excel file for an organized spreadsheet of the entire Google Ngrams data on the word jumbo. }} }
\label{table:JumboData}
\end{table}

From these data we have created a scatter plot of percentage of printed words that are the word jumbo versus time which is overlaid with an appropriate exponential function. 


\begin{figure}
\begin{center}
\includegraphics[width = 11 true cm]{1-32-JumboGrapha.eps}

\vspace*{-9.7 true cm}
\parbox{9.5 true cm}{\caption{A scatter plot of percentage of printed words that are the word jumbo versus time for the years 1880 to 1940. The red line is the best-fit exponential function, $P(t) = 5.07231223e^{.0549975t}$.}}
\end{center}
\vspace*{-.5 true cm}
\label{fig:1-32-JumboGrapha}
\end{figure}


The above graph is included to show the students that it is possible to find the best-fit exponential function, but is not intended to suggest that the students themselves find such a function at this point in the semester.  It is suggested that this additional exercise only be included if students have had prior experience finding functions to fit data.  If that has not been part of the curriculum, but will be a component of the differential equations course, the instructor could mention that the students will learn this skill at some point soon.

If the instructor's classroom has a computer/projector, then this is a good time to show the students other graphs from Google Ngram Viewer. Some interesting words to plot include groovy, ration, dude, and melodramatic. We have included screenshots of the graphs for groovy and ration below. If the instructor does not have the appropriate technology, then this page could also be printed as a handout.

Have a wrap-up discussion about these graphs and their implications. Lead back to a discussion about $k$ and why different words propagate through the literature at different rates. Also, it can be useful to discuss the limitations of the exponential model and picking an appropriate time scale.  

\subsection*{GROOVY}

Figure~\ref{fig:1-32-Groovy1a} shows the initial spread of the word groovy. 

\begin{figure}
\centering
\includegraphics[width = 13 true cm]{1-32-Groovy1a.eps}

\vspace*{-13.7 true cm}
\caption{Percentage of total printed words that is the word groovy from 1945 to 1971.}
\label{fig:1-32-Groovy1a}
\end{figure}


We notice that Figure \ref{fig:1-32-Groovy1a} looks somewhat exponential, and given our previous example we may want to model this with an exponential growth model. 

\begin{figure} 
\centering
\includegraphics[width = 13 true cm]{1-32-Groovy2a.eps}


\vspace*{-13.7 true cm}
\caption{Percentage of total printed words that is the word groovy from 1945 to 2008.}
\label{fig:1-32-Groovy2a}
\end{figure}

Considering a longer time scale, we see that groovy was an extremely trendy word for a short period of time. Thus, perhaps exponential growth would be an appropriate model for 1845-1971, but we could not use it to extrapolate beyond 1971. For this, we would likely need to research the etymology and history of the word groovy, and build a piece-wise model that fits both the data and the history. 


\subsection*{RATION}

Figure~\ref{fig:1-32-Ration1a} shows the initial spread of the word ration. 

\begin{figure} 
\centering
\includegraphics[width = 13 true cm]{1-32-Ration1a.eps}

\vspace*{-13.7 true cm}
\caption{Percentage of total printed words that is the word ration from 1880 to 1920.}
\label{fig:1-32-Ration1a}
\end{figure}




We notice that Figure~\ref{fig:1-32-Ration1a} looks somewhat exponential, and given our previous example we may want to model this with an exponential growth model. But first, let's consider a larger time scale. 

\begin{figure}
\centering
\includegraphics[width = 13 true cm]{1-32-Ration2a.eps}

\vspace*{-13.7 true cm}
\caption{Percentage of total printed words that is the word ration from 1880 to 2008.}
\label{fig:1-32-Ration2a}
\end{figure}

Zooming out shows that there are likely other factors influencing the propagation of the word ration. For example, we see two separate peaks, occurring after World Wars I and II, which would likely be motivated by real-world events and therefore not predicted by the same sort of exponential growth as with the word jumbo. 



\subsection*{Suggestions for Future Study and Student Projects}

Studying the propagation of different words through the verbiage is a rich area for students to have fun with differential equations. As the previous examples suggest, words have very different graphs of propagation, and investigating this further yields a wide range of student projects. The following is a list of possible extensions of the work in this activity that will require students to explore and experiment with the Google NGram viewer. 
\begin{itemize}
\item Students could find words that have exponential-looking graphs, but with different growth rates. They could then compare the growth rates and try to argue why particular words spread faster or slower than others. 
\item This project focused on exponential growth, and the next model students usually learn is logistic growth. Students could identify words that level off and try to build a logistic model to fit those data. We suggest the example of ``melodramatic" as it has an overall logistic-looking shape with detailed variations. 
\item Students could look at trends in word introductions during times of turmoil in our society, e.g. WWI, WWII, VietNam War, Great Depression. The example of ``ration" has peaks at WWI and WWII.
\item We considered the data for jumbo only through 1940 in an effort to highlight the part of the data that looked exponential; after 1940 the usage dies off a little until about 1960 when it picks back up. Students could use a piecewise function to model each stage the propagation for jumbo or other words. This idea speaks to one of the original learning outcomes for this project. Namely, a model that only works on a short time-scale is not necessarily ``wrong" but instead a researcher must choose a model (or part of a model) that best fits their needs and interest of study. 
\item The word jumbo is a 1gram because it is just one word, but Google Ngram Viewer has the capability of searching ngrams. Students could investigate the propagation of words that are commonly used together. For example, ``applied mathematics" or ``pure mathematics" as opposed to just ``mathematics." 
\item Depending on the interests of the students in the class they could investigate the spread of words belonging to specific categories. For example, they may consider business buzz words or scientific jargon. This could be a good place to interject the idea of using mathematics in social justice and investigate terms dealing with race, gender, or religion.  
\end{itemize}

}


\subsection*{Acknowledgments}  

We wish to thank Professor Michael Drout of Wheaton College for suggesting a project using Google Ngrams.  We also appreciate Giampaolo Di Gregorio's suggestion of historical references for the etymology of the word jumbo.  The authors also thank the referees for their thoughtful and helpful feedback on improving this project.


\begin{thebibliography}{98}

\bibitem{barnum} {\it Jumbo the Elephant}.  The Barnum Museum.  \href{https://www.barnummuseumexhibitions.org/jumbotheelephant.htm}{https:/www.barnummuseumexhibitions.org/jumbotheelephant.htm}.  Accessed  15 July 2015.

\bibitem{tufts} Sauer, Anne. {\it The Story of Jumbo}. Originally found in {\it Tufts Journal}, October, 2001.  Reprinted online at \href{http://tuftsjournal.tufts.edu/archive/2001/october/tufts150/index.shtml}{http://tuftsjournal.tufts.edu/archive/2001/october/tufts150/index.shtml}.  Accessed  15 July 2015.

\bibitem{google} Google Books Ngram Viewer.  \href{http://books.google.com/ngrams}{http://books.google.com/ngrams}. Accessed 15 July 2015.


%\bibitem{Barnum_interview} P.T. Barnum, interview, ``Philadelphia Press," April 22, 1882


\end{thebibliography}












\end{document}
