chap1-mathmodelling.tex

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		Chapter 1 - Mathematical Modelling
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{topic}[Mathematical Modelling]
%\Title{Mathematical Modelling}

\label{chap1}
In this section, we study some strategies to model problems mathematically in an effective manner.

We also provide a structure to modelling problems by breaking them in small parts:

\begin{enumerate}[label={\bf \arabic*.}]
	\item \hyperref[moddefine]{Define the problem}
	\item \hyperref[mindmap]{Build a mind map}
	\item \hyperref[assumption]{Make assumptions}
%	\item \hyperref[D-parvsvar]{Decide on your parameters and variables}
	\item \hyperref[model]{Construct a model}
	\item \hyperref[analysis]{Analyze the model}
	\item \hyperref[report]{Write a report}
\end{enumerate}

\vspace{1cm}

In this chapter, we follow the approach of 
\begin{graybox}
\begin{minipage}{.8\textwidth}
\begin{verbatim}
	Math Modeling: Getting Started and Getting Solutions, K. M. Bliss, 
	K. R. Fowler, and B. J. Galluzzo, SIAM, Philadelphia, 2014
\end{verbatim}
\begin{center}
\url{https://m3challenge.siam.org/resources/modeling-handbook}
\end{center}
\end{minipage}
\hfill
\begin{minipage}{.17\textwidth}
	\hfill\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}	
\end{minipage}
\end{graybox}
and
\begin{graybox}
\begin{minipage}{.8\textwidth}
\begin{verbatim}
	GAIMME: Guidelines for Assessment and Instruction in Mathematical
	Modeling Education, Second Edition, Sol Garfunkel and Michelle
	Montgomery, editors, COMAP and SIAM, Philadelphia, 2019
\end{verbatim}
\begin{center}
\url{http://uoft.me/gaimme}
\end{center}
\end{minipage}
\hfill
\begin{minipage}{.17\textwidth}
	\hfill\qrcode{http://uoft.me/gaimme}	
\end{minipage}
\end{graybox}


\vfill


\begin{center}
\begin{minipage}{300pt}
	\includegraphics*[width=300pt]{images/chap1-xkcd.jpg}

	\hfill {\footnotesize (image adapted from \href{https://www.xkcd.com/605/}{xkcd - comic \#605})}
\end{minipage}
\end{center}




\end{topic}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - DEFINING THE PROBLEM
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{module}{Defining the Problem}
	%\Title{Defining the Problem Statement}
	\label{moddefine}

	\input{modules/module01-define.tex}
	\input{modules/module01-define-exercises.tex}
\end{module}


\begin{lesson}
	\Title{Defining Problem Statement}

	
\Heading{Textbook}
	\begin{itemize}
		\item Module 1
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item The first step in Mathematical modelling is to define the problem
		\item A good way to do this is to figure out what is the ``mathematical object'' we are looking for at the end of the process

	\end{itemize}
	
\Heading{Motivation} 

\begin{annotation}
\begin{notes}
\begin{itemize}
	\item Students usually want to start thinking of ways to solve the problem.
	
	\item They need to learn to tackle a modelling task step-by-step.
\end{itemize}
\end{notes}	
\end{annotation}

Students have heard the words ``Model'' and ``Modelling'', yet they don't have a good idea what it means. 

The goal of this first chapter is to establish a common procedure to approach all  (or at least most) modelling tasks. 

The very first step in the procedure is to \emph{define the problem} in a clear and Mathematical way.

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}

\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 10-14} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
\end{lesson}






\def\email{
%	\thispagestyle{siam2019}
	\begin{graybox}
	-------- Forwarded Message -------- \\[10pt]
	\textbf{Date: } \dayofweekname{7}{9}{\the\year}, 7 September \the\year \; 21:41:35 + 0000  \\
	\textbf{From: } CEO <theCEO@theBigCompany.ca> \\
	\textbf{To: } Human Resources <hr@theBigCompany.ca> \\
	\textbf{Subject: } they're still late! \\
	
	Hey Shophika! \\
	
	I still get complaints about staff being late, some by 15 minutes.
	
	With the staff we have, that's about one salary lost.
	
	Again the bottleneck of the elevators seems to be the problem.
	
	Can you suggest solutions? \\
	
	Thanks, the CEO
	\end{graybox}
	
	(\newurl{problem adapted from GAIMME, SIAM}{http://uoft.me/gaimme})
}

%Elevator problem at theBigCompany

%\addcontentsline{toc}{subsection}{Task 1.A: Elevator problem at theBigCompany}

\begin{annotation}
	\begin{goals}
		\Goal{Make the question precise, bring it into a ``mathematical form''.}
		\begin{itemize}
			\item Choose a mathematical object best suited for the problem, e.g. a number, a geometric form, a graph, a function, an algorithm, ...

		\end{itemize}
	\end{goals}
%	\begin{notes}
%		
%		\begin{itemize}
%			\item There are many ways to solve this problem.
%				Some students
%				might start with equations. After they use their
%				equations to solve the problem, make them draw a picture
%				and come up with a graphical solution.
%
%			\item When the students start coming up with vector equations,
%				give them the vocabulary of \emph{linear
%				combinations}
%				and \emph{column vector notation}.
%		\end{itemize}
%	\end{notes}
	\begin{notes}
		
		\begin{itemize}
			\item Students will start discussing how to solve the problem
			\item This question deals with what will happen \textbf{after} solving the problem
			\item The goal of this question is to think about how to best tell a ``mathematically-challenged'' CEO that you solved the problem
			\item Student teamwork: ``With your team, you must decide on one answer and be prepared to report on your decision and the reason for your choice.''
		\end{itemize}
	\end{notes}
\end{annotation}

\question \label{elevator-define}
You are hired by theBigCompany to help with their ``elevator problem''.

This is the email you received:

\begin{center}
\begin{minipage}{.75\textwidth}
	\email
\end{minipage}
\end{center}


%\vspace{5mm}

What mathematical object would you use to convince the CEO that you have solved or improved the problem? 


\begin{teamwork}
	With your team, you must decide on \emph{one} answer and be prepared to report on your decision and the reason for your choice.	
\end{teamwork}

\bookonlynewpage







\question 

The mayor of Toronto wants to extend the subway line with a new \textbf{\color{orange}orange line} as in the figure below. \label{p:TTC}
	
%	\begin{center}
%	\includegraphics*[width=500pt]{images/TTC-extension.png}
%	\end{center}

\begin{center}
	\includegraphics*[width=500pt]{images/TTC.png}
\end{center}
\begin{minipage}{1.1\textwidth}
\hfill {\footnotesize (Map taken from \href{https://uoft.me/modelling-TTC}{Wikimedia Commons} created by Craftwerker) \qquad \qrcode[height=30pt]{https://uoft.me/modelling-TTC}}
\end{minipage}
	
\begin{parts}
	\item What ``mathematical object'' would you use to communicate that to the Mayor that this line is optimal (or sub optimal) ?

	\item Define the problem mathematically.
\end{parts}
















\standardonlynewpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - Mind Map
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{module}{Building a mind map}
	%\Title{Building a mind map}
	\label{mindmap}

	\input{modules/module02-mindmap.tex}
	\input{modules/module02-mindmap-exercises.tex}
\end{module}



\begin{lesson}
	\Title{Building a mind map}


\Heading{Textbook}
	\begin{itemize}
		\item Module 2
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
%		\item The second step is to create a mind map of the problem. This is a structured way to brainstorm possible solutions, their requirements, and other objects that affect or are affected.

		\item The second step in Mathematical modelling is to construct a representation of how the team will be attempting to solve the problem.

		\item Create a mind map of the problem. This is a structured way to brainstorm possible solutions, requirements, other objects that are related, etc.
	\end{itemize}
	
\Heading{Motivation} 

In this step, students are supposed to brainstorm and relate the problem at hand with everything that is affected or can be affected by it.

The idea is to get a simple visual representation of possible solutions, without all the details. \\

This is a fundamental step in modelling.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:dragons}: \dragonstitle
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}


\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 10-14} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
\end{lesson}




%\newpage


\question
\label{elevatorR}
Consider the elevator problem from question \ref{elevator-define}.


\begin{annotation}
	\begin{notes}
	\begin{itemize}
		\item Students usually come up with more complicated variations:
		\begin{itemize}
			\item Money spent on late employees' salaries
			\item sum of time in minutes that employees are late counting only employees that are at most 15 minutes late
		\end{itemize}
		\item Stick with $T$, a simple first approach
	\end{itemize}	
	\end{notes}
	
\end{annotation}
Your team decides that the mathematical object you will use to show the CEO that you solved or improved the problem is
\begin{itemize}
	\item $T=$ the sum in minutes by which every employee is late.
\end{itemize}

Note that employees that are on time count for 0 minutes (not a negative amount of minutes). \\

Create a mind map for the question: \quad How can $T$ be minimized?


\bookonlynewpage


\question

The city of Toronto decided to tear down the Gardiner expressway. While the demolition is taking place, several key arteries are closed and many intersections are bottled. 
At peak times, a police officer is often posted at this intersection to \emph{optimally} control the traffic lights. 

\begin{parts}
	\item What mathematical meaning can we give to the word optimal in this circumstance? 
	\item Create a mind map for this problem.
\end{parts}




	









\standardonlynewpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - Making Assumptions
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{module}{Making assumptions}
	%\Title{Making Assumptions}
	\label{assumption}

\begin{siam}
	\input{modules/module03-assumptions.tex}
	\input{modules/module03-assumptions-exercises.tex}
\end{siam}
\end{module}
	



\begin{lesson}
	\Title{Making Assumptions}

\Heading{Textbook} 
	\begin{itemize}
		\item Module 3
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item The third step is to decide on a path to the solution and start making assumptions.
		
		\item This is a difficult balance between:
		\begin{itemize}
			\item Accuracy, but difficult to analyze/solve;
			\item Simple, easy to analyze/solve but not very accurate.
		\end{itemize}

	\item Make sure assumptions and conditions of the modelling are clearly mentioned to the future reader/user of the model
	\end{itemize}
	
	
\Heading{Motivation} 

Students often make assumptions explicitly and implicitly, but they often keep them out of their notes. In the end they forget to include them in the final report.

\begin{annotation}
\begin{goals}
	\Goal{Assumptions.}
	Include the assumptions in the model's final report.
\end{goals}
\end{annotation}
It is imperative that they include their assumptions in the final report of their model.

Moreover, students should make an effort to find out the implicit assumptions and the conditions that to the model that their assumptions require.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
\end{itemize}

\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:dragons}: \dragonstitle
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}

\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 15-19} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
\end{lesson}




%\newpage

\begin{minipage}{.5\textwidth}	
\question
\label{elevator-assumptions}
Consider the elevator problem from \hyperref[elevator-define]{core exercise \ref{elevator-define}}. 


We now give you some technical details about \nobreak{theBigCompany}:

\begin{itemize}
	\item The company occupies the floors 30--33 of the building Place Ville-Marie in Montr\'eal.

	\item Personnel is distributed in the following way: 
	\begin{itemize}
		\item 350 employees in floor 30,
		\item 350 employees in floor 31,
		\item 250 employees in floor 32, 
		\item 150 employees in floor 33.
	\end{itemize}
\end{itemize}

\emph{Note.} Even though these details are fictional, the numbers respect the building code. \\

\emph{Hint.} Focus on a \textbf{few} parameters and variables.
\end{minipage}
\qquad
\begin{minipage}{.5\textwidth}	
\email
\end{minipage}

\begin{annotation}
	\begin{notes}
		\begin{itemize}
			\item Students usually have trouble starting. 
			\item They usually agree that they have to figure out how elevators work, so you can prompt them to be more specific. 
			
			\item In the end they should come up with questions like these:
			\begin{itemize}
				\item How fast are the elevators?
				\item How much time do elevators take in each floor?
				\item How many floors do elevators stop on their way up?
				\item How many people fit in the elevator?
				\item Should we consider elevator failures?
			\end{itemize}
		\end{itemize}	
	\end{notes}
\end{annotation}
\begin{parts} 
	\item With your team, decide on what kind of information you would need to have to be able to solve this problem.

	\item Find the relevant information about the elevators (search the internet, by experimentation). Check the reliability of the data you found.

	\item For the relevant information that you cannot obtain, make assumptions. These assumptions should be reasonable and you should be able to justify them.
\end{parts}




\bookonlynewpage

\begin{notslides}

\hfill

\bookonlynewpage
\end{notslides}

\question How much would it cost to make a bridge between Toronto and the U.S.?














\standardonlynewpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - Construction of the Model
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{module}{Construct a model}
%	\Title{Construct a model}
	\label{model}

\begin{siam}

In this module you will learn
\begin{itemize}
	\item how to build a model based on the previous steps
\end{itemize}

\hfill

	\input{modules/module04-model.tex}
	\input{modules/module04-model-exercises.tex}
\end{siam}

\end{module}



\begin{lesson}
	\Title{Construct a model}

\Heading{Textbook} 
	\begin{itemize}
		\item Module 4
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item The fourth step is to use the mind map created in step 2, the assumptions from step 3, and assemble everything into one model.
		
		\item The model is not the solution to the problem: it is the framework to solve the problem. 
	\end{itemize}
	

\Heading{Motivation} 

\begin{annotation}
\begin{notes}
	\begin{itemize}
	\item Model $\neq$ Solution
	\end{itemize}
\end{notes}
\end{annotation}
Students usually think that the solution is the model that they need.

Emphasize that students should see the example in the textbook to get an idea of what a model looks like.

\begin{annotation}
\begin{notes}
	There won't be enough time in class to finish the question if students don't prepare in advance.
\end{notes}	
\end{annotation}

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Ask students to prepare the core exercise \ref{model:preclass} before class.

	\item In class, they should combine their mind map(s) and assumptions from the previous lessons and come up with an idea of a model to discuss with their classmates in lecture.

\end{itemize}

\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:dragons}: \dragonstitle
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}


\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 20-31} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
\end{lesson}




%\newpage



\begin{minipage}{.5\textwidth}	
\question \label{model:preclass}
Recall the \hyperref[elevator-assumptions]{core exercise \ref{elevator-assumptions}}.

\begin{itemize}
	\item The company occupies the floors 30--33 of the building Place Ville-Marie in Montr\'eal.

	\item Personnel is distributed in the following way: 
	\begin{itemize}
		\item 350 employees in floor 30,
		\item 350 employees in floor 31,
		\item 250 employees in floor 32, 
		\item 150 employees in floor 33.
	\end{itemize}
\end{itemize}

%\vspace{1 .5cm}

Write down a mathematical model for this problem.
\label{elevator-model}

\begin{teamwork}
Each team should have \emph{one} model and be prepared to present it to the class.	
\end{teamwork}

\end{minipage}
\qquad
\begin{minipage}{.5\textwidth}	
\email
\end{minipage}

\begin{annotation}
\begin{goals}
	\Goal{Create a model.}
	\begin{enumerate}[label=\textbf{\color{Cerulean}\arabic*.}]
		\item Students should join in teams of 2--4 and come up with \emph{one model} for the team.
		\item The model should include:
		\begin{itemize}
			\item Definition of the problem
			\item Mind map
			\item Assumptions and conditions
			\item Clearly defined path to solve the problem
		\end{itemize}
		\item A few teams present their model for everyone else.
		\item Full class brainstorm about each model:
		\begin{itemize}
			\item Does it include all the parts specified?
			\item Is it solvable? 
			\item Could we make some extra assumptions to make it simpler?
			\item Is it accurate?
			\item Could we change something to make it more accurate without sacrificing simplicity?
		\end{itemize}
	\end{enumerate}
\end{goals}	
\begin{notes}
Suggestions to help students get started:
\begin{itemize}
	\item How long do all employees take to get to work?
	\item When do they arrive to the building?
	\item First: evaluate the situation -- don't try to fix the problem
\end{itemize}	
\end{notes}
\end{annotation}


















\standardonlynewpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - Model Assessment
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{module}{Model Assessment}
	%\Title{Model Assessment}
	\label{analysis}

\begin{siam}
	\input{modules/module05-assessment.tex}
	\input{modules/module05-assessment-exercises.tex}
\end{siam}

\end{module}




\begin{lesson}
	\Title{Model Assessment}

\Heading{Textbook} 
	\begin{itemize}
		\item Module 5
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item After creating a model, students should make an analysis of the model
		\item The analysis is meant to test the model as well as obtain some of its consequences
		\item If there is a solution to the model, here is where it can be found and analyzed
	\end{itemize}
	

\Heading{Motivation} 

Once a model is created, one must check if the model solves the problem it is meant to solve.
There are different types of assessment for a model:
\begin{itemize}
	\item Check some known cases to see if it works as expected;
	\item Check some extreme cases to see if it works as expected;
	\item Check some implications of the model to make sure they are reasonable;
	\item Check that the assumptions made are reasonable for the problem at hand;
	\item If possible, use approximation methods to estimate the solution;
	\item If possible, find the solution and analyze it.
\end{itemize}


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:dragons}: \dragonstitle
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}


\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 32-39} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
\end{lesson}




%\newpage

\begin{annotation}
\begin{notes}
	Some questions to guide the students:
	\begin{itemize}
		\item What are the strengths of this model?
		\item What are the weaknesses of this model?
		\item Is the result around what you expected?
	\end{itemize}	
	
	\hfill \\
	In case students don't realize that something is wrong:
	\begin{itemize}
		\item People start arriving 30 minutes before the starting time, so \emph{almost everybody will be on time?}
		\item Assume that the CEO of the BigCompany is right: people are arriving late! What's wrong with the model?
	
		\item Which assumptions should be relaxed? Or checked?
		\item If one needs to be replaced, by what?
		\item Do we need extra assumptions? Which?
	\end{itemize}
\end{notes}
\end{annotation}
\question

Continuing on the \hyperref[elevator-model]{elevator problem}, let us think of this model for the problem.

\textbf{Facts:}
\begin{itemize}
	\item Loading time of people at ground floor = 20 s
	\item Speed of uninterrupted ascent/descent = 1.5 floors/s
	\item Stop time at a floor = 7 s
	\item Number of elevators serving floors 30--33 = 8

	(these elevators serve floors 23-33 = 11 floors)
	
	\item Maximal capacity of elevators = 25 people
\end{itemize}


\textbf{Assumptions:}
\begin{itemize}
	\item Personnel that should start at time $t$, arrive uniformly in the interval $[t-30, t-5]$ in minutes
	\item First arrived, first served
	\item During morning rush hour, elevators don't stop on the way down
	\item Elevators stop only at half the floors they serve
	\item Elevator failures are neglected
	\item Mean number of people per floor is equal to the mean number of people per floor of the BigCompany
	\item Elevators are filled, in average, to 80\% of their capacity
\end{itemize}

\begin{annotation}
\begin{goals}
	\Goal{First make sure model works. Then try to find a solution.}
	Notice that the model doesn't attempt to find a solution to the question. \\
	
	If there weren't any problems with this model, we could then start asking other questions:
	\begin{itemize}
		\item How can we get people in their office faster? 
		\item How will each idea affect the estimate?
		\item Will they cost money to the company?
	\end{itemize}
\end{goals}
\end{annotation}
\slidesonlynewpage
\questionagain
\textbf{Model:}
\begin{itemize}
	\item Mean number of people per floor $= d = \dfrac{350+350+250+150}{4} = 275$ people / floor
	\item Number of people on floors served by elevators (11 floors) $= N = d \cdot 11 = 3025$ people
	\item Time $\Delta t$ of one trip

\hfil $\Delta t \quad = \quad $ \framebox{$\substack{\text{loading time on}\\\text{ground floor}}$} 
		$ \;+ \;$ \framebox{$\substack{\text{time of flight}\\\text{ground $\to 33$}}$}
		$ \;+\; $ \framebox{$\substack{\text{time of flight}\\\text{$33 \to$ ground}}$}
		$\;+\; $ \framebox{$\substack{\text{stop time to}\\\text{6 of the 11 floors}}$} $\quad = \quad$ 106 s
		
		\item Number of trips necessary per elevator $= n = \dfrac{3025}{20 \cdot 8} \approx 19$ trips

		\item Time necessary to carry the staff of the BigCompany $= \pmb{t} = \dfrac{19 \cdot 106}{60} = 33 $ minutes
		
		\item Accumulated late time $ = \pmb{T} = 180 \cdot 20 \cdot 8 + 74 \cdot 20 \cdot 8 = 40\,640$ seconds $= $ 11h18m

\end{itemize}

\hfill


Your task is to assess this ``model'' (first estimate of the number of minutes employees are late). Be ready to report on your assessment.

\begin{teamwork}
Each team should have \emph{one} assessment and be prepared to present it to the class.	
\end{teamwork}



















\standardonlynewpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MODULE - Report
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{module}{Putting it all together}
	%\Title{Putting it all together}
	\label{report}

	\input{modules/module06-report.tex}
	\input{modules/module06-report-noexercises.tex}
\end{module}



\begin{lesson}
	\Title{Putting it all together}

\Heading{Textbook} 
	\begin{itemize}
		\item Module 6
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Students practice writing a ``technical'' report
	\end{itemize}
	

\Heading{Motivation} 

Writing technical reports requires practice and students haven't had to do this before.


Some time can be devoted to this in lecture or tutorial, but if that's the case, then students must write a short report before the class and bring it with them. 


\begin{annotation}
\begin{goals}
	\Goal{PAR}
	\qrcode{https://link.springer.com/article/10.1007/s40753-015-0005-y}
\end{goals}	
\end{annotation}
In class, the instructor can use the \href{https://link.springer.com/article/10.1007/s40753-015-0005-y}{Peer-Assisted Reflection (PAR)} format of Daniel Reinholz as in the example in the next two pages:
\begin{itemize}
	\item Students bring their report (1 page);
	\item Students self assess their own work: students check off the icons for things you think you did well and circle the icons for things they would like feedback on;
	\item In class students exchange reports and write feedback to each other;
	\item Students go home and revise their report and submit it online.
\end{itemize}

\emph{Note:} Students are also not practiced at giving good feedback, so although the PAR rubric helps mitigate this, we recommend that the instructor spends some lecture/tutorial time showing an example of a question and different answers where students can practice giving good feedback.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook;
	\item In teams of 2--4, write a short report on a modelling problem (e.g. see next two pages).
\end{itemize}


\Heading{Projects}
\begin{itemize}
	\item Project \ref{proj:dragons}: \dragonstitle
	\item Project \ref{proj:erdos}: \erdostitle
	\item Project \ref{proj:elephants}: \elephantstitle
\end{itemize}


\begin{annotation}
	\vspace{-20pt}
	\begin{goals}
		\Goal{$\text{M}^{\text{2}} (\text{GS})^{\text{2}}$}
		\qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
	\end{goals}
	\begin{goals}
		\Goal{GAIMME}
		\qrcode{http://uoft.me/gaimme}
	\end{goals}
\end{annotation}
	\Heading{Extra Reading} 
	
	\begin{itemize}
		\item \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions ($M^2(GS)^2$), Bliss-Fowler-Galluzzo -- pages 40-44} \\
		\hfill \url{https://m3challenge.siam.org/resources/modeling-handbook}
		
		\item \href{http://uoft.me/gaimme}{Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), Garfunkel-Montgomery, editors}
		\hfill \url{http://uoft.me/gaimme}
	\end{itemize}
%\end{lesson}


%\begin{instructoronly}

\newpage

\input{example/PAR.tex}
	
%\end{instructoronly}
\end{lesson}





%\newpage

%\question
%Another question to be added here
%
%\newpage





%
%
%
%
%\begin{module}
%	\Title{Putting it all together}
%	\Heading{Textbook} \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions, Bliss-Fowler-Galluzzo}
%	
%	\Heading{Objectives}
%	\begin{itemize}
%		\item Bla bla bla	
%	\end{itemize}
%	
%	\Heading{Motivation} 
%
%
%\end{module}
%
%
%
%
%\section*{Step F. Writing a report}\label{F-report}
%\addcontentsline{toc}{subsection}{Step F. Writing a report}
%