Math 319: Math Modeling
Summer Semester 2010
Note: this syllabus is temporary, and may change up to the first day of class.
This version posted on: 2010-06-02
Math Modeling is the art of taking a real-world problem and stating it
in mathematical terms. It often involves making simplifying assumptions.
In our class, we get in the habit of doing all the parts of the
math modeling cycle: modeling, solving, checking, and guessing.
Often, a large part of the problem is even deciding which problem to solve.
For example, should you find the best schedule for your staff at one
location, or consider opening new locations? Should you
start with a theoretical model then match it to data, or just model
the data directly? We will also consider a lot of common
mathematical models, and explore their properties.
Course Catalog Entry
The modeling process; model building and evaluation, techniques of
modeling; model fitting and models requiring optimization; empirical
model construction---experimental models, dimensional analysis, simulation
models, dynamic models; use of derivatives in the modeling process,
single and multivariable dynamic models.
Math 120 and Math 122.
Some experience using Excel, VBA, Mathematica, Maple, or Matlab will also
be VERY helpful, but it is not strictly a prerequisite.
Follow-up courses: Math 325 Differential Equations, Math 418 Modeling with Linear Algebra, Math 419 Advanced Math Modeling (stochastics), Math 425 Math for Scientists, Math 436 Numerical Analysis
The U of M has two related courses: Math 462, "Mathematical Modeling", and Math 463, "Mathematical Modeling in Biology". However, these
focus on differential equation models, while this class focuses on regression, operations research, and dynamical systems.
Mon, Tue, Thu. 1:00-2:50pm in Mark-Jefferson 122
"Final Exam" schedule: Tue, Aug. 17th, usual class time
CRN 41956, 3 credit hours.
Class meetings will be mostly interactive lectures, with some time to work on
problems in class, and some time to discuss homework.
Professor Andrew Ross
(734) 487-1064, but I strongly prefer e-mail instead of phone contact.
Math department main office: Hoyt 404, (734) 487-1444
Office Hours and other help
Office Hours: tentatively
- M/T/R 9:30am-10:30am (location TBA)
- M/T/R 12:20-1:00pm (starting at Roosevelt 101, walking to MarkJ 122, eating lunch on the way)
- M/T/R 3:00-4:00pm (location TBA)
I am also happy to make appointments if you cannot come to the general
office hours. Please send me e-mail to arrange an appointment.
I am definitely unavailable during the times I teach other classes:
The Mathematics Student Services Center (or "Math Lab") is also here to
help you, in
117A King Hall.
Their hours are posted here.
Please give them a call at 734-487-0983 or just drop by.
Many assignments in this course will be in the form of papers, which I
want to be well written. Please consult with
The Writing Center
for help in tuning up your writing.
Most students do well in this course without a textbook. For those who
feel the need to have one just in case, I suggest
"A First Course in Mathematical Modeling", 3rd or 4th Edition, by
Giordano, Weir, and Fox.
A lot of our work will be done on computers, specifically in Excel. If you
had been waiting for a good reason to buy a laptop, this is it.
Course Web Pages
I will post data files, homework assignment files, etc. on
my home page.
We will use an on-line gradebook (perhaps the
WebCT system) to keep track of grades.
You are expected to keep an eye
on your scores using the system, and get extra help if your scores
indicate the need.
Here is a list of books that I have found interesting and related to
math modeling. Perhaps some of them will strike your fancy, too.
I own the ones that are starred (*) and can lend them to you.
Others you will have to find at the library or on the usual Internet
booksellers. Links are given to Amazon, but I do not specifically endorse
them or any particular bookseller. Of course, if you like a book you can
see what similar books the online bookseller recommends.
Here are some journals that you might be interested in:
- *Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics by Robert B. Banks, and its sequel:
- *Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics by Robert B. Banks
- Race car vehicle dynamics by William F. Milliken, Douglas L. Milliken
- Going Faster: Mastering the Art of Race Driving by Carl Lopez
- Logistics Systems Analysis by Carlos Daganzo
- The nature of mathematical modeling by Neil Gershenfeld
- Mathematical and Experimental Modeling of Physical and Biological Processes by Banks and Tran
- *Small is Profitable by Amory B. Lovins, et al.
- Statistical Orbit Determination by Tapley and Schultz
- Street-Fighting Mathematics (free online book!)
- Quantitative Methods in Health Care Management: Techniques and Applications, by Yasar A. Ozcan (EMU library)
- Patient Flow: Reducing Delay in Healthcare Delivery, edited by Handolph W. Hall (EMU Library, incl. electronic copy)
- Mathematical Modeling for the Life Sciences, by Jacques Istas
- Turning Numbers into Knowledge: Mastering the Art of Problem Solving
by Jonathan G. Koomey (EMU library, incl. electronic copy)
- *The Active Modeler: Mathematical Modeling with Microsoft Excel by Erich Neuwirth
- * Mathematical Modeling With Excel by Brian Albright
- Methods of Operations Research (1951) by Morse and Kimball--especially for military applications
- Military operations research (1997) By N. K. Jaiswal
- Applied operations research (1988) By Ronald William Shephard--especially for military applications
- Microsoft Excel, or other spreadsheet software like Gnumeric or OpenOffice or Google Docs
- Mathematica, Maple, or Matlab/Octave/Scilab
Our primary goal is to teach you to be a good (or great!) math modeler.
To be a good modeler, you need:
We have a few secondary goals, which may be more or less applicable to your
- Good habits and procedures, just like a scientist, and
- Knowledge of common math models.
- Get enough people together to form a few teams for the
Math Contest in Modeling (MCM), mid-February, 2011. I participated in this
3 times as an undergrad and had a lot of fun. Recent EMU teams have done well!
- Give future teachers some great ideas to
show your kids how high-power math is used in the real world. You may enjoy reading Meaningful Math.
- Give computer-science students lots of interesting things to program. You may like reading this blog entry about math for programmers.
- Get everyone using Mathematica, Maple, or Matlab/SciLab.
- Teach you how to communicate your math models by writing math papers and giving math presentations.
By the end of the course, students will be able to:
- (General modeling skills):
- categorize problems into operational/tactical/strategic categories,
- identify nearby problems in the oper./tact./strat. hierarchy,
- evaluate models by constructing simple test cases,
- conduct cross-validation when needed,
- select the most important variables to start modeling with,
- (Empirical modeling skills):
- use ordinary, semilog, and loglog plots to evaluate relationships in data sets,
- perform linear regression in software,
- interpret the correlation coefficient,
- perform transformations before regression as appropriate,
- fit a function to data using nonlinear regression, e.g. sine waves
- decide when to use logistic regression (logit), and interpret the results
- (Communication skills):
- write a technical report,
- differentiate between literature of varying quality, e.g. peer-reviewed vs. working paper vs. white paper vs. web site,
- design appropriate figures to communicate models and results,
- (Optimization skills):
- Formulate non-linear programs (NLP) as appropriate,
- solve NLP using software,
- describe the (im)possibility of multiple optimal solutions (convexity/concavity)
- formulate linear programs (LP) as appropriate,
- solve LP using software,
- describe the nature of LP solutions,
- identify common LP models: network flow, diet/blending, inventory, minimax, assignment,
- formulate Integer programs (IP) as appropriate,
- identify common IP models: knapsack, fixed charge, scheduling
- (Dynamical Systems skills):
- use Dynamical Systems to model populations:
- single-variable, incl. financial models,
- two-population: predator/prey, competition, cooperation
- age-structured populations (Leslie models)
- Markov-chain models
- Fit model parameters to data,
- Describe equilibrium behavior,
- Implement and interpret trajectory plots, phase-plane plots, and delta-a_n versus a_n plots
- (Other models):
- describe basic Queueing models,
- describe the Traveling Salesperson problem (TSP)
- describe project-scheduling models (PERT)
- describe dynamic-programming models (DP)
Here we show which chapters from the book we will probably cover,
in roughly the order we will cover them. A star (*)
denotes full coverage, a plus (+) denotes partial coverage, and no
symbol denotes no coverage. For example, DTMCs (as cool as they are)
will be covered in Math 419 rather than 319.
Ch 2:+ proportionality, similarity
Ch 3:* model fitting, least-squares
Ch 4:+ experimental modeling, high-order polynom, low-order polynom, splines
Ch 5:+ simulation
Ch 6: Discrete Time Markov Chains (DTMCs)
Ch 8:+ modeling using graph theory
Ch 7:+ Linear Programming (LP), one-dim. line search
(and add Integer Programming?)
Ch 13:* Non-Linear Programming (NLP), inventory
Ch 9:+ dimensional analysis and similitude
Ch 10: graphs of functions as models
Ch 1:* difference equations, dynamical systems
Ch 11:+ one-dim ODEs
Ch 12:+ systems of ODEs
Some variations in this outline are to be expected.
Regular attendance is strongly recommended. There will be material
presented in class that is not in the textbook, yet will be very useful.
Similarly, there are things in the textbook that
are might not be covered in class, but are still very useful.
If you must miss a class,
arrange to get a copy of the notes from someone, and arrange for
someone to ask your questions for you.
My lectures and discussions mostly use the chalkboard, along with
demonstrations in Excel and other mathematical software. I do not
usually have PowerPoint-like presentations, and thus cannot hand
out copies of slides.
Homework will be assigned about once every week. It will
sometimes be a small problem set designed to help you
understand the behavior of math models. Other times, it will involve
writing up a little paper on an assigned topic. All homework should
Homework papers should be submitted on-line, where they might be
TurnItIn.com or a similar
service. This is partly to help keep you honest, and partly to help you
learn acceptable ways to cite the work of others. A side benefit is that
sometimes TurnItIn finds papers relevant to your work that you would not
have found otherwise!
There will be no exams, unless the class demonstrates an unwillingness
to be motivated any other way.
Instead of a mid-term and a final exam, you will do a mid-term and a final
project. Your results will be reported in a paper and a presentation to
the class. You may work by yourself or in a team of 2 people, but no
groups larger than 2 will be allowed. You may switch project partners
at your will. Your project grades will each be split something like this:
The final presentations will be made during the
time slot reserved for the final exam.
- 10 pct: proposal
- 80 pct: work and written report
- 10 pct: presentation
On average, students should spend a total of about 30 minutes in office hours
discussing the project. Plan for this in advance!
No scores will be dropped, unless a valid medical excuse with evidence
In the unfortunate event of a medical need, the appropriate grade or
grades will be dropped entirely, rather than giving a make-up.
You are highly encouraged to still complete the relevant assignments and consult with me during office hours to ensure you know the material.
Your final score will be computed as follows:
Final percentage scores will be given letter grades as follows:
- 50 percent for all the homework together,
- 20 percent for the mid-term project, and
- 30 percent for the final project.
- 92.0 and above : A
- 88.0 to 92.0: A-
- 84.0 to 88.0: B+
- 80.0 to 84.0: B
- 76.0 to 80.0: B-
- 72.0 to 76.0: C+, etc.
The instructor reserves the right to make changes to this syllabus
throughout the semester. Notification will be given in class or
by e-mail or both. If you miss class, it is your responsibility
to find out about syllabus and schedule changes, especially
the due dates and times of projects, assignments, or presentations.
Advice from Other Math Modeling Students
In the last two semesters, I've asked my math modeling students to give
advice to you, future math modeling students, based on their experiences
in my course. Here are some of the highlights:
* work in groups * start the first day assignment is given * don't take too many credits w/ this class * ask a lot of questions * utilize Dr. Ross
Do go to his office hours more than you normally would; if you have a question ask don't wait.
See Prof. Ross in office hours and don't be afraid to email him. He is usually very helpful and approachable.
Plan on visiting Prof. Ross during office hours in order to do well in the class. You will learn a lot in the end, but be ready to work.
- [prof ross:] add a note to the syllabus stating something to the effect of, "This class will not be like other math classes. Instead of straight-up problems or proofs, the biggest amount of work will be setting up the models, exercises, etc. and in analysing what your results mean. It will not be the mathematical work done to obtain the results that is the tricky part." But word the note better.
- attend the office hours Prof Ross is really good at explaining & helping out with the homework
- WORK TOGETHER!
- Take notes during the computer lab days and send yourself the excel sheets.
- Go to class. The computer lab days help even if you know excel well.
- Go to class. Go to office hours and pick project that you're energized about and interested in even if they're harder. It will make this math class the best one you've ever taken.
- Don't drop the class! It sounds impossible in the beginning, but stick with it.
- Don't procrastinate.
- Take differential equations close to this class, it will make more sense!
- Start projects ASAP.
- Ask questions!!! The professor will guide you along the way like Yoda.
- Talking to anyone about your projects or the homework, be it Prof. Ross or other students, is a really, really good idea.
- Never be afraid to ask for help.
- If project falls through, have backups.
Schedule for Projects
Thu Jul 15: Proposal 1 due
Tue Jul 27: Project 1 due; presentations start
Thu Aug 5: Proposal 2 due
Tue Aug 17: Project 2 due; Presentation 2 due; presentation day!
Standard University Policies
Religious Holy Days
I support students' right to observe religious holidays without
penalty. To the best of my ability, I will schedule exams to not conflict
with major religions' holidays. Students are to provide advance notice to the instructor in order to make up work, including examinations that they miss as a result of their absence from class due to observance of religious holidays. If satisfactory arrangements cannot be made, the student may appeal to the head of the department.
Academic dishonesty, including all forms of cheating and/or plagiarism, will not be tolerated in this class. Penalties for an act of academic dishonesty may range from receiving a failing grade for a particular assignment to receiving a failing grade for the entire course. In addition, you may be referred to the Office of Student Judicial Services for discipline that can result in either a suspension or permanent dismissal. The Student Conduct Code contains detailed definitions of what constitutes academic dishonesty, but if you are not sure about whether something you’re doing would be considered academic dishonesty, consult with the instructor.
Students are expected to abide by the Student Conduct Code and assist in creating an environment that is conducive to learning and protects the rights of all members of the University community. Incivility and disruptive behavior will not be tolerated and may result in a request to leave class and referral to the Office of Student Judicial Services (SJS) for discipline. Examples of inappropriate classroom conduct include repeatedly arriving late to class, using a cellular telephone, or talking while others are speaking. You may access the Code online at
Special Needs Accomodations
If you wish to be accommodated for your disability, EMU Board of Regents policy #8.3 requires that you first register with the
Access Services Office (ASO) in room 203 King Hall. You may contact ASO by telephone at (734) 487-2470. Students with disabilities are encouraged to register with ASO promptly as you will only be accommodated from the date you register with them forward. No retroactive accommodations are possible.
Student and Exchange VISitors (SEVIS)
The Student Exchange Visitor Information System (SEVIS) requires F and J students to report the following to the
Office of International Students, 229 King Hall within ten (10) days of the event:
Prior permission from OIS is needed for the following:
- Changes in your name, local address, major field of study, or source of funding.
- Changes in your degree-completion date
- Changes in your degree-level (ex. Bachelors to Masters)
- Intent to transfer to another school
Failure to report may result in the termination of your SEVIS record and even arrest and deportation. If you have questions or concerns, contact the OIS at 487-3116, not your instructor. Also, see the
EMU SEVIS page.
- Dropping ALL courses as well as carrying or dropping BELOW minimum credit hours
- Employment on or off-campus
- Registering for more than one ONLINE course per term (F-visa only)
- Endorsing I-20 or DS-2019 for re-entry into the USA