Math 319: Math Modeling

Prof. Andrew Ross

Fall 2016

Basic Information

Note: this syllabus is temporary, and may change up to the first day of class.
This version posted on: 2016-09-02

General Description

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 419W Introduction to Stochastic Mathematical Modeling, 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.

Class Meetings

Section 0, CRN 11189: Mon/Wed 11:00-12:15 in Pray-Harrold 521; 3 credit hours
Detailed schedule:
Prof. Andrew Ross
Math 319, CRN 11189;
Mon/Wed 11:00-12:15, Pray-H 521
Block#Date 2016dayunittopicsBonus Tech before classHW assignedHW due
19/7Wedgeneral modelingintro; math model examples; a math model has; graph sketchingM1
29/12Mongeneral modelingbloom's taxonomy; CCSS-M standards for mathematical practice; malaria nets--start simple; evacuation; modeling cycletext-to-columnsM2M1
39/14Wedgeneral modelingreal modeling cycle; oper tact strat; airline problems; concept maps; intro to excel (graphing, label axes, title, autofill, control-shift-down)left/mid/right and =DATEM3M2
49/19Monregressionlinear regression: houses, predictions, residuals, graph residuals!vlookupR1, R2M3
59/21WedregressionR^2; school district data; correlation/causation; ecological fallacy; common resid graphs; basic procedure; LSRL math model; averaging before regression?marked scatterplotsR3R1
69/26MonregressionPre-Lab at home: 4-function pre-quiz; in-class: answers; exponential fits, compound interestsparklinesR3 before class, R2
79/28Wedregressionyeast; logplots; power fitPivot TablesR4
810/3Monregressionlog-of-log, model selection, occam's razor, multivariate regression school dataparallel axis plotsR5R4
910/5Wedregressionheat index; polynom; sinesLiveRegressionR5
1010/ java fourier app; waves and trendsR6
1110/12WedregressionQuiz on R5; Logistic; overfitting/crossvalidation; Machine Learning overviewgenerating random numbersR7, R8, R9R6
1210/17MonoptimizationLP toys, wyndor (no sensitivity analysis), knapsack, swimmersO1R7
1310/19Wedoptimizationshift scheduling; network flowO2R9
1510/26WedoptimizationMCNF node-node; ramen; brief fast-food intro; sensitivity analysis on wyndor; feas region; fundamental theorem of LPO3, M4O2
1610/31Monoptimizationexample papers: dinosaur and relay; NLP: manufacturing, electricityPasting into Word/PPT: live or dead copies?O3, M4
1711/2Wedoptimizationconcavityproposal 1
2011/14MondynsysDynamical Systems; PID; credit card, repeated dosingO4, O5
2111/16WedprojectsProject Presentationsreport 1; presentation 1
2211/21MonprojectsProject Presentations
2311/23WedThanksgiving break
2411/28Monlimited population growth; quizD1
2511/30Weddynsyspagerank; leslie; SIR; pred/prey;oilspillD2D1
2612/5Mondynsysmultiple initial conditions; equilibria; delta plots; phase-plane plots; fitting limited-pop growthD3proposal2
2712/7Weddynsysobservation noise, process noiseD2
2812/12Mondynsysrepeated dosing? accel/vel/pos? chaos? splines? PERT/CPM?D4, M5, M6D3
2912/14Wedwrapup; M5 & M6 discussionD4, M5, M6
12/19Monpresentationspresent in "final exam" slot: 11:00am (usual class time)report 2; presentation 2

Class meetings will be mostly interactive lectures, with some time to work on problems in class, and some time to go over problems from the homework.

I expect that you will work on Math 319 for 6 to 9 hours per week outside of class during a regular (Fall or Winter) semester, and 2.5 times that during a Summer semester (6-week session)

Instructor information

Professor Andrew Ross
Pray-Harrold 515m
(734) 487-1658, but I strongly prefer e-mail instead of phone contact.
Math department main office: Pray-Harrold 515, (734) 487-1444

Office Hours and other help

Here is my complete schedule.
 10:00-11:00 Office Hours
 11:00-12:15 Math 319, PH 521
 12:30- 1:45 Stat 360-0, PH 305
  1:45- 2:45 Office Hours
  2:45- 3:15 (Wed) faculty research meeting
 11:30-12:30 Office Hours
 12:30- 1:45 Stat 360-1, PH 305
  1:45- 2:45 Office Hours 
  4:30- 5:30 Office Hours
  5:30- 6:45 Math 560, PH 503
	no schedule--I'm often on campus, though.
	I have various meetings to go to.
	Send e-mail to make an appointment.

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.

The Mathematics Student Services Center (or "Math Lab") is also here to help you, in Pray-Harrold 411 Their hours are posted here. Please give them a call at 734-487-0983 or just drop by.

A good place to study, if the Math Lab doesn't suit you, is the Math Den, Pray-Harrold room 501.

Another resource on campus is the Holman Success Center, formerly the Holman Learning Center.

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.

Teaching philosophy, interests

I am a very applied mathematician. Applied, applied, applied. Not pure. Impure. I try to focus on real-world problems, rather than artificial drill problems (though I do recognize the need for some drill). My classes spend much more time on formulating problems (going from the real world to math notation and back) than on proving theorems. If you want the theoretical basis for anything we are discussing, please ask!

My general math interests are in Industrial Engineering and Operations Research (IEOR). In particular, I do research in applied probability and queueing theory, the mathematics of predicting how long it takes to wait in line for service. You can learn more about this in Math 319 and 419 when I teach them. I also enjoy teaching about cost-minimizing/profit-maximizing methods called Non-Linear Programming (NLP) in Math 560.

(not-absolutely-)Required materials

Most students do well in this course without a textbook. For those who feel the need to have one just in case, I suggest finding "A First Course in Mathematical Modeling", any edition, by Giordano, Weir, and Fox, in a library or the Math Den (PH 501).

A lot of our work will be done on computers, specifically in Excel or other spreadsheet software (except Apple Numbers). If you had been waiting for a good reason to buy a laptop, this is it. Spreadsheets other than Excel (such as OpenOffice/LibreOffice, Google Docs, etc.) work reasonably well for most things in the class, but some things really don't work well without name-brand Excel. Fortunately, it's available free to EMU students (as of Fall 2016). Email me to ask for details.

Course Web Pages

I will post data files, homework assignment files, etc. on my home page.

We will use an on-line gradebook (via EMU Canvas) 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. Nearly everything will be submitted via the various dropboxes inside EMU Canvas. The rule is: if it's not in a dropbox, it doesn't exist (for grading purposes).

Supplementary Materials

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: Other Stuff:

Course Content

Course Goals

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 personal situation:

Student Outcomes

By the end of the course, students will be able to: (optional topics that we might not get to are marked with a ?) Also compare this list of outcomes to the CUPM 2015 course guide for math modeling.


This course was originally organized around the Giordano modeling textbook, though it is not required for the course. Here we show which chapters from that book we 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.

Grading Policies


Regular attendance is strongly recommended. Since there is no formal textbook, missing class means you will miss a lot! 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 document camera, 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 per class meeting, though some assignments are short. 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 be typed unless noted.

Homework papers should be submitted on-line, where they might be checked by TurnItIn 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.

On average, students should spend a total of about 30 minutes in office hours discussing the project. Plan for this in advance!

Overall Grades

In the unfortunate event of a need, the appropriate grade or grades might be dropped entirely, rather than giving a make-up, at the instructor's discretion. 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:

General Caveat

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:

Standard University Policies

In addition to the articulated course specific policies and expectations, students are responsible for understanding all applicable University guidelines, policies, and procedures. The EMU Student Handbook is the primary resource provided to students to ensure that they have access to all university policies, support resources, and student's rights and responsibilities. Changes may be made to the EMU Student Handbook whenever necessary, and shall be effective immediately, and/or as of the date on which a policy is formally adopted, and/or on the date specified in the amendment. Please note: Electing not to access the link provided below does not absolve a student of responsibility. For questions about any university policy, procedure, practice, or resource, please contact the Office of the Ombuds: 248 Student Center, 734.487.0074,, or visit the website: CLICK HERE to access the University Course Policies