During tests, Dr. Barzilai has a pile of scratch-paper available for students. This allows them to have have a place for scratch-work which they do not turn in, in case they need it, before writing the neat solutions on their tests. Students are told they are to turn in only their tests, not their scratch-paper, and reminded they can keep and recycle that paper. Dr. B. notices many students nevertheless throw their scratch-paper into the trash basket when they leave the room upon completing and handing in their tests, and wonders how many sheets of paper which could have been recycled, are thrown away this way each semester in this classes, or in all SU classes.

1. Based on the above assumptions, create a model (this is part i.) and then give an estimate (this is part ii.) for the number of pieces of paper thrown in the trash, per semester, in Dr. B.'s classes, assuming also 4 courses taught per semester, and an average of 3 tests plus a final, so 4 exams in all, per course.

Part i: A Model. Each semester there are 4 classes Dr. B teaches, so to get an idea of how many test-takers there are we can multiply this by some number S representing the average number of students per course, with each one taking 2 pieces of paper for each of 4 exams. This means that each semester 4S test-takers will each use 2*4=8 pieces of paper. So multiplying the number of test-takers per semester, by the number of sheets of scratch-paper each uses during the semester, we get 4S · 8 or 32S sheets of paper, in this model. Part ii: An Estimate. A google search for "Average class size site:salisbury.edu" suggests an average class size of 27. However that includes "large lecture hall" courses, while the Mathematics Dept. at SU teaches in classrooms with a max of 30 or so chairs, so we chose to use 22 for S instead, as an average of small upper level classes, and freshmen classes with up to 30 or so students. Thus 32S becomes 32·22 or 704 sheets of paper. To give the reader a better feel or visual, a standard ream of paper has 500 sheets, so roughly, or just under, one and a half reams of paper per semester are thrown in the trash, according to this model and estimate.

2. Create a model for, and come up with an estimate for the number of pieces of paper thrown away and not recycled counting all classes at SU (not just Dr. B.'s), per semester. (You may want to google "2007 FTE site:salisbury.edu" or the like, where FTE stands for Full-time Equivalent students, but not all models will need an FTE in them)

A search suggests that for Spring 2007 there were 6,444 FTEs (Full-time Equivalent) students at SU, based on www.salisbury.edu/president/pat/20070208%20PAT%20February%208,%202007%20minutes.DOC

We need to know or to estimate how many courses an SU student is enrolled in, on average, each semester. We will use the figure of 5.5 courses per semester. Multiplying gives us (6,444)·5.5 = 35,442 for the number of times a student is enrolled in a course (per semester).

However, since these are for Spring 2007, not Spring 2008, let's assume (without having the exact numbers in front of us) a 5% growth took place in FTEs between Spring 2007 and Spring 2008 (the above .DOC file mentioned a 5.4% increase had taken place to reach those Spring 2007 numbers. We are assuming the growth rate for SU continues at close to that rate). Thus multiply 1.05 by 35,442 to get about 37,214 as the number of times any student is enrolled in any class (during Spring 2008).

An alternative model: use http://www.salisbury.edu/registrar/courseschedules.html to find the course schedule for spring 2008 http://www.salisbury.edu/registrar/Schedules/Spring%2008,%20Jan%2011.pdf to estimate the number of courses (counting each section of a course separately) offered each semester. Rather than hand-count, we can either call the Registrar (thus jamming their phone if each student in this class does this) or use the above pdf to make an estimate. The PDF has 82 pages. One random page, page 18, has 33 sections listed On page 47 picked at random, we happen to reach the Math Department and with two or more lines per section listed and only 26 sections listed. We could average a larger number of random pages to get an more accurate estimate, but using just these two numbers, we will use for our model an average of (33+26)/2 = 29.5 courses (sections) per page, times 82 pages which would be 2,419 sections for that semester. However there is some blank space in this listing, so we will round down by 10% to 2,200 sections, which we will assume is a typical semester in this regard. Using 27 students on average, per course/section, we obtain (27* 2,200) = 59,400 for the number of times a student is enrolled in a course (per semester). This is somewhat higher than the earlier model which gave about 37,000, but it is in the rough general ballpark (more than half, and less than twice, of our earlier number, in this case) This gives us confidence that we are probably not too far off.

For individual homeworks, and certainly for the course project, if you are able to calculate or estimate something two ways, that gives you a way to check the reasonableness (or not) of your calculation or estimate. (Project reports whose teams employed innovative ways of calculating or estimating a number in two or more ways will be evaluated more highly).

Here we will take the average of the two numbers: (37,214 + 59,400)/2 = 48,307 or roughly 48,000 as our estimate of how many times any student sits in any course/section (During Spring 2008). If we assume each such student takes 4 exams, including the final, then about 4·48000 or about 192,000 exams are taken each semester.

Modeling can be an "Art" as much as a "Science" and several questions arise: how many of those exams involve students using scratch paper? How much scratch paper? Probably many exams do not use scratch-paper we it might be an over-estimate to multiply the above number by 2 pieces of paper per student. On the other hand, we might have an under-estimate of wasted paper (even paper used for academic purposes alone, not paper cups etc). For example, we did not count the exams themselves, which the professors may either recycle or throw away after a holding period of a year or so. More significantly, what did we forget to count as far as paper used for academics?

We have not counted paper used by students for class-notes (how much gets kept, for how long? How much is recycled? How much thrown in the trash and landfilled, etc? In addition, we have not counted student essays (times a certain number of pages per essay), much less all the drafts which students print out before printing out the final draft). This second set of considerations suggests the ways in which multiplying just the number 48,000 times the number of pieces of paper, if any, thrown away as scratch work, might be an under-estimate.

A more precise "answer" (i.e., a more precise estimate) would take a project beyond the scope of one homework assignment, but just to complete the exercise, let us ignore both the above ways in which we might over-estimate and how we might under-estimate waste of paper used for academic purposes, and multiply "192,000 exams taken" by 2 sheets of paper hypothetically thrown into the trash to arrive at a little over 384,000 sheets of paper, or 768 reams of paper, per semester, thrown into the trash instead of recycled. Do you think this is an over-estimate or under-estimate, overall? By a little? By a lot?

For questions 1 and 2, What assumptions or estimates are you making? What might cause the actual number to be higher? To be lower? (come ready to comment on this in the class discussion).

Extra Credit: very roughly, how many pieces of paper are thrown away at U.S. universities per year? What assumptions or estimates are you making? For this problem (question 2.), what are your assumptions and estimates, and what might cause the actual number to be lower or higher? Also, how much higher would the figure be if we counted not just paper used for academic purposes, but also other kinds of paper (paper cups etc) which is thrown away rather than recycled? What kinds of variables would this add to the model? How might you estimate their values? (come ready to discuss in class, or share insights in your PEMJ) P.S. Don't sweat over whether your numbers are exactly correct. Don't worry about a perfect answer....