A World Wide Web Project
Department of Mathematics
University of Scranton
Scranton, PA 18510
Department of Mathematical Sciences
Allentown, PA 18104
You are caught in bumper-to-bumper traffic heading south to downtown Chicago on Lake Shore Drive. Tuning your radio to the traffic station, you grit your teeth as you hear that your normal fifteen-minute commute time from Montrose Street to Randolph Street has been replaced by forty minutes of torture. With all that time on your hands, you start wondering: "How do they calculate traffic times in Chicago?"
While most people think of helicopters circling high above the traffic, the Illinois Department of Transportation's Traffic Systems Center (TSC) uses a different approach. The TSC receives its data from loop detectors embedded in the expressways in the Chicagoland area. The loop detectors act like metal detectors and can sense when a vehicle passes over them. This allows the TSC to count the number of vehicles that have passed over each detector as well as how long each detector was occupied. The TSC probably uses a simple application of Riemann sums (or better numerical approximations such as Simpson's rule) to convert this data into travel times and congestion estimates. The data is updated every minute. A more extensive explanation of the system can be found at http://www.ai.eecs.uic.edu/GCM/GCM.html.
We have incorporated this data into a writing project suitable for first-year calculus students studying Riemann sums. Written in the form of a mystery, the project cannot be solved without some mathematical detective work. In an attempt to exonerate the suspect, students must calculate approximate travel times using numerical approximations. They also must present and explain their solution to someone with a limited mathematical background (in this case a lawyer and the jury). In addition to applying calculus to real-world data, this project gives students the opportunity to improve their ability to communicate a mathematical solution to a non-mathematical audience, a task which forces them to evaluate the depth of their understanding of these concepts.
What follows is the writing project written in the form a letter, comments about the project, sample data, and a map.
Barney, Smith and Elmo
100 Lower Wacker Drive
Chicago, IL 60601
June 1, 2000
100 Main Street
College Town, USA 45678
Dear Calculus students:
I am a lawyer for the firm of Barney, Smith and Elmo. One of my clients has been accused of robbing the downtown Chicago branch of MegaCityBank on January 27th of 2000. My client claims that he left work and went directly home to study for an exam in his evening class. The police claim that he robbed the bank on his way home. Allegedly, my client bears some resemblance to the thief caught in the act by the bank cameras. I need to convince a jury that my client did not commit this crime, but I can't do this without your help. I have gathered a lot of evidence concerning the events of the 27th, but I don't know how to piece it all together. I haven't had a mathematics course in years and, although I'm embarrassed to admit that I can't solve this puzzle, I'm willing to ask for help.
Allow me to explain the events of January 27th. A validated parking ticket with an exit time of 1:44PM stamped on it shows when my client left the parking garage in his building. The bank robbery took place at exactly 2:02PM as shown by the video cameras at the bank. Since the parking garage clock runs a minute behind the bank clock, my client had 17 minutes to get from his building to the bank. MegaCityBank is located just off of the Dan Ryan Expressway at Roosevelt Road, right in the heart of downtown Chicago (which is what Chicagoans call "the Loop"). My client works south of the Loop, and the Dan Ryan Expressway is the quickest possible route from his workplace to the Loop. It takes my client about one minute to enter the Dan Ryan Expressway at 95th Street and head northbound towards the Loop, where MegaCityBank is located. As part of my client's defense, I want to show that travel times on the Dan Ryan were such that it would have been impossible for him to commit the crime. Right now you're probably thinking that securing this type of information is nearly impossible. Fortunately, this isn't the case. The WorldWideWeb and the Illinois Department of Transportation (IDOT) have come to our rescue.
The Gary-Chicago-Milwaukee Corridor Transportation Information Center provides expressway traffic information. In fact, it gives travel times to and from the Loop for all of the major expressways in Chicago. The Center's computer receives raw data from the IDOT Traffic Systems Center (TSC) once every minute. The TSC receives its data from loop detectors embedded in the expressways. The loop detectors act like metal detectors and can sense when a vehicle passes over them. This allows the TSC to approximate the speed of traffic at each loop detector. Simple formulas have been developed to convert this data into travel times and congestion estimates. A more extensive explanation of the system can be found on their web site at http://www.ai.eecs.uic.edu/GCM/GCM.html.
I have included a copy of the data from the Dan Ryan Expressway Loop Detectors from January 27th at 1:45PM. I am interested in determining the northbound travel time from 95th Street to Roosevelt Road using the data from the detectors. Now, to complicate matters, the Dan Ryan has both express lanes and local lanes and the travel times for these are typically not the same. Heading northbound the expressway splits into express and local lanes at 65th Street. So, after entering the Dan Ryan at 95th Street my client could have used either the local or the express lanes starting at 65th Street. Also, travelers can only exit at Roosevelt Road from the express lanes. So, if he did use the local lanes then he would have to switch back to the express lanes by 29th Street in order to be able to exit at Roosevelt Road. So, I am also interested in determining the travel time if my client took the northbound local lanes from 65th Street to 29th Street and used the expressway lanes for the rest of the trip. I don't know if this will be helpful, but I drove the Dan Ryan from 95th Street to Roosevelt Road and the distance is approximately 10.4 miles. The express and local lanes are just separated by a concrete median, so the distance is the same for either. I have also included a map of the relevant region.
As I told you above, although I took calculus in college many years ago, my knowledge of mathematics is limited. Please include a clear and thorough explanation of how you arrived at your answer. Be sure to include and explain any formulas that you may have used. I have to understand your solution completely in order to explain it to a jury. Thanks for your help.
G.E. Toffelhook, Esquire
Here is a table of sample data for the twenty-one detectors between 95th Street and Roosevelt Road. This allows students to use twenty intervals in their calculations. To simplify calculations, we suggest that students assume the detectors are more or less evenly spaced. To add another level of complexity to the problem, one could delete the distance that the lawyer includes at the end of the letter, and instead tell students that eight blocks are roughly equal to one mile, Root Street is equivalent to 41.5th street, and Roosevelt Road is equivalent to 12th street. Using this, students would not need to assume that the detectors are evenly spaced. It is also worthwhile noting that the detectors do not register speeds in excess of 55 mph. Students might explore what possible variation in travel times would occur if they considered a maximum speed of 60 or 65 mph for those detectors which indicate 55 mph. For this sample data, the accused would have to travel in excess of a ridiculous 500 mph in order to complete the expressway portion of his trip in 20 minutes. Here is a map to help students visualize the trip.
We have assigned this project to five second-semester calculus classes at two different schools. We have assigned it both individually and in groups of three or fewer students. Here are some of our observations. While the travel time the students' calculate is only an estimate, some students correctly point out that they are only given the speeds at one particular moment in time and not at several different times and hence, their estimate is not as accurate as it could be. More commonly, students make the mistake of incorrectly calculating an "average" speed by adding up the speeds and dividing by 21, then dividing the distance by this rate to calculate the travel time. Typically, we have already warned them about this mistake by asking them in class to compute the average velocity of a 20 mile trip where the driver's speed is 30 mph for the first 10 miles, but 60 mph for the last 10 miles. It is possible to manipulate the sample data such that when calculated incorrectly, the travel times will be small enough as to not be helpful to the lawyer, but when calculated correctly, the travel times exonerate the client. If one wishes to update the data, the northbound Dan Ryan is most congested in the mornings and early afternoons. The data is located at http://www.ai.eecs.uic.edu/GCM/dan-ryan.html.
As with any writing project, this project takes students' time and effort to produce good results. It is essential to emphasize that the students' ability to explain their reasoning is just as important as their calculations, and will factor heavily in their grade. To encourage this, we suggest that it be assigned at least two weeks before its due date and that it be worth more than other homework, perhaps 5% of their final grade. To grade the project we use a checklist similar to those used by Annalisa Crannell at Franklin and Marshall College (A. Crannell, PRIMUS 4(3):1994).