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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Drone Heading Home}


\markboth{Drone Heading Home}{Drone Heading Home}

\author{Richard Spindler\\
SUNY Plattsburgh\\
Plattsburgh NY 12901 USA\\
rspin004@plattsburgh.edu
}

\makeStitlePDFLaTex


\begin{abstract}
We describe a comprehensive project in modeling through a two dimensional system of equations and a first order system of differential equations. Students are guided in creating the model in a collaborative, inquiry-based manner but independently analyze it analytically, qualitatively, and numerically. The project contains diverse opportunities for extensions, including analyzing parameter sensitivity and using it in a variety of applied pursuit curve scenarios. The rubric used to guide the project report writing and to evaluate the project is tied to outcomes in the modified Bloom's taxonomy. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}
You just received a new long-range helicopter drone for your birthday! After a little practice, you try a long-range test of it by having it carry a small package to your home. A friend volunteers to take it 5 miles east of your home with the goal of flying directly back to your home. So you program and guide the drone to always head directly toward home at a speed of 6 miles per hour. However, a wind is blowing from the south at a steady 4 miles per hour. The drone, though, always attempts to head directly home. We will assume the drone always flies at the same height. What is the drone’s flight path? Does it get the package to your home? What happens if the speeds are different? What if the initial distance is different? How much time does the drone's battery have to last to get home?


In this project, you will have two submittals:

\begin{enumerate}
\item \textbf{Draft Report:} After you collaboratively work on developing your model(s) of this scenario in two class sessions, you will submit your introduction to the problem, as well as your draft differential equation model(s). The purpose of this submittal is to ensure you have an accurate model before moving forward and also to receive feedback on that model. See the rubric regarding the introduction and the model description. 
\item \textbf{Final Report:} After receiving feedback on your model(s) from the instructor, you will submit a final report where you mathematically analyze the model, make predictions, interpret your predictions and conclusions, evaluate those conclusions, and reflect on the project experience. See the rubric for the detailed requirements. Here are some pertinent questions that provide specific ideas for prediction that you should address and you may have more of your own to add: 
\begin{itemize}
\item What is the drone’s flight path? 
\item Does the drone get the package to your home? 
\item Can you obtain an estimate of how long the drone’s battery has to last to get home? How might you attempt that? 
\end{itemize}
``What if'' questions: Consider some of these “what if” questions and how they affect the pertinent questions:
\begin{itemize}
\item What if the wind or drone speeds are different? 
\item What if the initial distance from home is different? 
\item Do you have other reasonable questions?
\end{itemize}
Do any of those ``what if''questions affect battery or other aspects of drone design? 
\end{enumerate}
Be sure to utilize the rubric as your guide in writing your final report. 

\teacher{
\section*{NOTES FOR TEACHERS}

\bigskip

This project is filled with learning experiences where students must draw on trigonometry, vector calculus, specific techniques in differential equations, and physical intuition. The idea for this pursuit curve model originated from and is similar to a pursuit curve model as applied to geese \cite{Borrelli 2004}. It can be found in older differential equations books such as \cite{Agnew 1960} and variations of it prior to appearing in differential equations books.  

\bigskip

Students first work on building the model for this scenario in two class sessions each of length 15 to 30 minutes. There are several inquiry-based and active learning models you may use to facilitate this.  How much and how detailed the guidance is will depend on how much scaffolding the particular students require. Some teaching models to consider are: 
\begin{itemize}
\item The instructor acts as a facilitator with the entire class asking questions at the appropriate time.
\item Students work collaboratively in groups to work through progressive guidance given by the instructor.
\item Students work individually (or possibly in groups) outside of class to work through written guidance given by the instructor. Students could then present their results in class or put results on classroom boards on which other students comment on, such as a gallery walk. 
\end{itemize}

\bigskip

Here is a general outline we used to guide the class in creating the draft model(s) for the draft report. Note that it starts with a smart problem-solving strategy, which is to draw a picture. In the spirit of inquiry-based learning, typical questions or directions we might ask or give the students are given in quotes.
\begin{enumerate}
\item ``What coordinate system should we use? Where do you want the origin (0,0) to be in the problem? What will be north/south/east/west?'' The students tend to choose home to be the origin (0,0) without any guidance. 
\item ``What do you want to name the variables for the position of the drone parameterized by time. What are important quantities you should name?'' Students then negotiate a little so we all use the same symbols. We also prod their memories. ``Can you recall parameterizations from Calculus 3? Who can describe what parameterizations are?'' 
\item We next give them some direction. ``Draw a picture of the problem, including a possible path of the drone. You should draw a path that makes intuitive sense to you. Also pick a general point on the path and a velocity vector for the drone at that position. Do you remember how to draw velocity vectors on a parameterized curve from Calculus 3? Who can show what a velocity vector looks like on a position curve?'' 
\item Then we give some advice: ``Later, it will be easier to have parameters (letters) for the drone speed and the wind speed rather than the numbers provided. At the end of your analysis, you can then experiment with various values. Or, if you are more comfortable with numbers at first, you can work on the problem with the given numbers and later generalize.''
\item Now we give a hint: ``To determine a system of equations, suppose the wind suddenly stops. Sketch the direction of the velocity vector of the drone at a representative position immediately after the wind stops.''  Many of them will be able to sketch the velocity vector heading directly toward home. The next step tends to be the difficult part of building the model for many of the students. 
\begin{enumerate}
\item To help them generate ideas from the picture, ``In your picture, identify two similar right triangles: a larger triangle with the legs of the triangle being components of the position vector of the drone and a smaller triangle with the legs being the two components of the velocity.'' The picture looks like that given in Figure \ref{fig:6-023-Model_Diagram}.

\bigskip

\begin{figure} [h] 
\centerline{\includegraphics*[angle=0,width=4in]{6-023-PursuitModelGraph.jpg}}
\caption{Drone Diagram} \label{fig:6-023-Model_Diagram}
\end{figure}

\item To obtain an initial pair of equations, ``Write the velocity components in terms of the speed and an appropriate angle (unknown right now) using trigonometry.'' It may be appropriate to discuss here how unit vectors are being used along with the magnitude given by the speed. 
\item Now ask: ``How can we determine the trigonometric functions of that unknown angle? What else in this picture can we use?'' If necessary, ask what information the larger right triangle contains. 
\item Next, ``Can you use that information from the larger right triangle so that you have a system of differential equations in terms of just the position components?''
\item Finally, ``Now suppose the wind suddenly starts again. How would you modify the system of equations?'' They should now have a reasonable first order system of equations model of the scenario. 
\begin{eqnarray}  \label{eq:SystemDE}
\frac{\ud x}{\ud t} & = & \frac{-bx}{\sqrt{x^2 + y^2}} \\
\frac{\ud y}{\ud t} & = & \frac{-by}{\sqrt{x^2 + y^2}} + w 
\end{eqnarray}
where 
\begin{itemize}
\item $(x(t),y(t))$ is the coordinate of the drone at time $t$;  
\item $w$ is the wind speed from the south (in the problem given as 4 miles per hour); 
\item $b$ is the drone speed  (in the problem given as 6 miles per hour);  and
\item the initial condition is $(x(0), y(0)) = (x_0, 0)$ (with $x_0 = 5$ miles east of home). 
\end{itemize}
\end{enumerate}
\item To derive a first order single equation of the model, ask ``What topic or result from Calculus 3 can we use to combine the two first order equations of $dx/dt$ and $dy/dt$ into one equation to obtain a first order single differential equation of $dy/dx$?'' 
The single differential equation is:
\begin{equation} \label{eq:SingleDE}
\frac{\ud y}{\ud x}  =   \frac{\frac{\ud y}{\ud t}}{\frac{\ud x}{\ud t}} = \frac{-by + w\sqrt{x^2 + y^2}}{-bx} = \frac{y}{x} - \frac{w}{b}\sqrt{1 + \Big(\frac{y}{x}\Big)^2}\ , \qquad y(5) = 0.
\end{equation}
\end{enumerate}

The students are given three to five days to to write up and submit an introduction to the problem and their understanding of the development of the model with 
explanations of the development. 
This initial draft submittal is worth 6\% of the total project grade, a relatively small percentage, and we give plenty of guidance and hints if  needed. 

In addition, if they have not yet encountered the type of procedure required to solve the first order differential equation, a homogeneous differential equation of the zeroth order, then some guidance or instruction through an example is given. 
To learn how to manage this type of differential equation, we first request students perform a change of variables to an example problem such as this
\begin{equation}\label{eq:DEPractice}
\frac{\ud y}{\ud x} = \frac{y - 4x}{x-y}
\end{equation}
providing guidance on the change of variables. Homogeneous differential equations of the zeroth order such as equations (\ref{eq:DEPractice}) and  (\ref{eq:SingleDE}) are amenable to to the substitution $u = y/x$, which transforms those equations into separable equations. (This guidance may not be necessary if students have been taught this substitution separately already.)   In this practice problem, we do ask students to do and submit their work on the substitution to verify they understand it. We do not ask students to finish the integration after the substitution unless we feel they need practice with integration techniques. The trigonometric substitution $u=4 \sec(v)$ is useful in the integration of (\ref{eq:DEPractice}) and $u=\tan(v)$ for (\ref{eq:SingleDE}). 

We give feedback quickly  to the students, within 2 days, on the draft submittal. 
In the second stage culminating in the final paper submittal, the students analyze the model(s) using techniques
learned in the course following a rubric. The rubric, provided in the Appendix, consists of the following elements: 
\begin{itemize}
\item \textbf{Model development}, improved or corrected if applicable, with explanations of the development. 
\item \textbf{Mathematical analysis} of the model(s) analytically, qualitatively, and numerically. 
\begin{itemize}
\item For the analytical analysis portion of this project, we allowed students to use Computer Assisted Algebra systems, such as WolframAlpha \url{http://www.wolframalpha.com/} or Mathematica, to solve the differential equation (\ref{eq:SingleDE}) after changing variables, but a couple of students solved it with paper and pencil. See the Appendix for an outline of the derivation. 
\item For the qualitative analysis, students have a number of options including phase lines and phase portraits. 
\item For the numerical analysis, students implement Euler's formula for either the single variable differential equation or for the system. 
\end{itemize}
This area of the rubric also entails comparing and contrasting the results from the different solution techniques. 
\item \textbf{Predictions}, including those requested of the student in the directions. 
\item \textbf{Interpretations} of predictions and conclusions, which means to discuss those predictions in terms of the applied problem about the drone, not simply providing the mathematics.
\item \textbf{Evaluation} of those predictions and conclusions, which means to critique and discuss the importance of the interpretations and predictions. 
\item \textbf{Reflection} on the project experience. This requests the student to look back on the process to ponder what was learned, mathematically or non-mathematically (such as work habits or attitudes) and what connections were made by doing the project. It also requests the student to look forward toward other possible investigations that could arise from this project. 
\end{itemize}
These elements were developed to map assessment with the modified Bloom's taxonomy of cognitive knowledge $\lbrack$2, 4$\rbrack$.
%\cite{Anderson et al. 2001} or  \cite{Krathwohl 2002}. 

To conclude, this project comprehensively assesses students on broad knowledge of techniques and concepts in a realistic applied context. 
Through the rubric, it also assesses deeper knowledge elements, including interpreting and evaluating. 
From the reflections it is clear these projects have a large impact on students. 
One student said it answered a big question: ``why am I learning about differential equations?''  
Some students commented on how much was learned in the project or how it improved their memory of the content. 
One student decided to add physics as a major based on project experiences, while another student is considering
adding mathematics as a major. 
In addition, some students said project work made them realize they need to plan better when doing projects. 
Finally, the project can be adjusted for different levels of challenge, depending on the needs of the student. 
In fact, one student has continued examining this problem as an undergraduate research project. 

The editor has provided an analysis of the system of differential equations (\ref{eq:SystemDE}) in a Mathematica notebook 6-023-T-Mma-DroneHeadingHome-TeacherVersion.nb and a pdf version of the same file for those who do not use Mathematica as well as an mp4 animation, 6-023-DroneHeadingHome.mp4,  produced in the above Mathematica notebook. These are found  in the Additional Materials for this Modeling Scenario.
\begin{thebibliography}{99}

%Chicago Manual of Style ?
\bibitem{Agnew 1960} Agnew R. 1960. {\it Differential Equations}. Hoboken: McGraw-Hill. pp. 146-147.

\bibitem{Anderson et al. 2001}Anderson, L. W. and D. R. Krathwohl 2001. {\it A taxonomy for learning, teaching, and assessing: A revision of Bloom's taxonomy of educational objectives}. New York: Longman.

\bibitem{Borrelli 2004} Borelli, R. 2004. {\it Differential Equations: A Modeling Perspective}. Hoboken: Wiley. pp. 145-148.

\bibitem{Krathwohl 2002} Krathwohl, David R. 2002. A Revision of Bloom's Taxonomy: An Overview. {\it Theory Into Practice}. 41(4):212-264. 

\end{thebibliography}

\section*{APPENDIX}
\centerline{\textbf{OUTLINE OF THE ANALYTICAL SOLUTION}}

\bigskip

For the analytical portion of the project, where students attempt to analytically solve the differential equation, students on their own (or with their project partner) apply that same substitution $u=y/x$ as in example equation (\ref{eq:DEPractice}) to equation (\ref{eq:SingleDE}) to obtain
\begin{equation} \label{eq:SingleDETransformed}
\frac{\ud u}{\ud x}  =  - \frac{k\sqrt{1+u^2}}{x}\, ,  \quad u(5)=0
\end{equation}
The wind parameters enter as a ratio,$ w/b$, and so here we define $k:= w/b$. 
(Some students did recognize this and defined some letter for it.) 
Most students have had enough experience to recognize this equation as separable, and so accurately write equation (\ref{eq:SingleDETransformed}) as
\begin{equation} \label{eq:SingleDESeparable}
\frac{1}{\sqrt{1+u^2}} \cdot \frac{\ud u}{\ud x} =  - \frac{k}{x}\, ,  \quad u(5)=0
\end{equation}
As mentioned in the main part of this article, we encouraged students to use Computer Assisted Algebra systems, such as WolframAlpha \url{http://www.wolframalpha.com/} or Mathematica, to solve this, but 
a few students solved it with paper and pencil using the trigonmetric substitution $u=\tan(v)$. With this substitution we obtain
\begin{equation} \label{eq:SingleDESolution}
u = \frac{1}{2} \Big\lbrack \Big( \frac{x}{5} \Big)^{-k} - \Big( \frac{x}{5} \Big)^{k} \Big\rbrack\, , \quad x \ne 0
\end{equation}
and then returning to the original variables, the solution is:
\begin{equation} \label{eq:SingleDESolution}
y = \frac{x}{2} \Big\lbrack \Big( \frac{x}{5} \Big)^{-(w/b)} - \Big( \frac{x}{5} \Big)^{(w/b)} \Big\rbrack\, , \quad x \ne 0
\end{equation}
Notice how the initial condition clearly displays itself in the final equation, which makes it easily varied as a parameter also. 
}

\vspace*{.4 true cm}

\centerline{\textbf{RUBRIC FOR THE FINAL REPORT} }

\bigskip

Each of the following elements are rated along a range of absent to excellent, depending on the completeness and quality of the work on that element.

\begin{description}
\item \textbf{Organization:} You must have the following organizational elements in the final paper, each with a clear header.
\begin{itemize}
\item Cover Sheet (title of your project, your name, your partners' names (if any) and the date)
\item Problem Introduction Section
\item Model Section
\item Mathematical Analysis Section
\item Prediction and Evaluation Section
\item Reflection Section
\item Credits: Give credit to people that assisted you or references you used (if any)
\end{itemize}

\item \textbf{Layout and Formatting:} This is a report, not an assignment, so please write this paper in narrative (paragraph) format, except putting equations and formulas on separate lines. Here are further guidelines:
\begin{itemize}
\item Narratives in paragraphs. Typed (except mathematical equations which may be hand-written.) 11 or 12 point font. 
\item Equations and calculations, etc. are on a separate line and easy to read. 
\item Understandability of graphs. (Suggestion: You may take a picture of hand drawn graphs and of graphs on calculators or other software.)
\item Clear identification of graphs in the narratives through labeling of the graphs (e.g., ``see Figure 1''). 
\item Graphs are used effectively in the narratives and explanations. In other words, you refer to the graphs and use the graphs in your discussions.
\end{itemize}

\item \textbf{Introduction:} Must include: 
\begin{itemize}
\item Introduction of the problem. (Summarize the problem in your own words.) 
\item 	Discuss how this problem may be similar and different from other applied problems you have encountered. 
\end{itemize}
 
\item \textbf{Model:}  Must include: 
\begin{itemize}
\item Clear mathematical statement of the model.
\item 	Definitions of variables. 
\item 	Explanation of the different parts of the model and how they relate to each other.
\item 	Justification of the model and its parts.
\end{itemize}



\item \textbf{Mathematical Analysis:} Must include:
\begin{itemize}
\item Qualitative Analysis: Discuss what mathematical information the qualitative analysis gives about the solution. 
\item 	Analytical Analysis: Solve and discuss what mathematical information the analytical analysis gives about the solution.
\item 	Numerical analysis: Solve and discuss what mathematical information the numerical analysis gives about the solution.
\item 	Explain: Each analysis should include an explanation of the steps taken analyzing the model. (But note you don't have to state or show every single basic algebraic step.)
\item 	Model verification: Are the results of the model consistent with any data? How? If there is no data, what else can you do to verify if the model results are accurate? 
\item 	Compare/Contrast: Compare and contrast the three techniques (graphical, analytical, numerical). What is similar and different? Did they agree with each other? For this problem, in what ways were they useful or not? 
\end{itemize}

\item \textbf{Prediction, Interpretation, and Evaluation:}  Must include: 
\begin{itemize}
\item Prediction: Make predictions based on the model. See the project directions. 
\item 	Interpretations: Be sure to state what your solution and predictions mean or say in terms of the applied problem. This should be done throughout the write-up as necessary. 
\item 	Evaluation: Critique the interpretations and predictions. Do the interpretations of the solution and predictions make sense for that applied problem? Why or why not? 
\item 	Evaluation: What might your solutions and predictions imply and why is it important for the people, animals, or objects in the applied problem? Does it imply any actions that should be taken in the future, i.e. do you have any recommendations based on your conclusion? 
\end{itemize}

\item \textbf{Reflection:}  Each person on the project must include separately a reflection with these elements:
\begin{itemize}
\item What was the most important thing you learned mathematically or about solving problems and did you learn anything mathematically new?
\item 	What was the most important thing you learned personally and did you learn anything new about yourself? (This could be about how you go about your work, how you communicate, your relationship with others, what you enjoy doing, your attitude towards mathematics, etc.)
\item 	Did you find the project difficult? In what ways? 
\item 	Did you find any connections to other disciplines or to courses you have had? 
\item 	Did you encounter any technical or non-technical problems? 
\item 	Did you encounter or think of other ways to attack the problem? 
\item 	Did other mathematical or physical questions arise that would be interesting to investigate? 
\end{itemize}
\end{description}

 

\end{document}
