Homework: Modeling and Estimation

Math 190e -- Week 1 Homework
Your assignments for this week are:

Turn in a Photocopy of your ID (or any other clear photo with your face) next class.
Review class activities and prepare (or create first draft) for the Week 1 Entry for your Professional Expository Mathematics Journal (PEMJ). See the handout on PEMJ for details. Your entry covers highlights (does not have to cover 100%) but you should draw from both class activities and presentations, as well as readings/HW, not just one of the other.
The Exploring the Google Calculator assignment (on a separate page). This should only take a half hour or so, a bit longer if you try the bonus question. This is mostly to familiarize yourself with the tools you need to maximize the effectiveness of all your work in this course and particularly your end of semester project/presentation (as well as insights you come up with on your own and put into your PEMJ)
The Modeling and Estimation below is the main HW assignment. We will discuss as a class next week and sample solutions will be handed out as well.

Recall that homework exercises are not to be turned in, with the exception of several designated assignments over the semester in which it will be announced that the assignment will be collected on a given date. The purpose is to help you deepen your conceptual understanding as well as to familiarize you with some skills. These will allow you to perform on both the mini-tests and to have high-quality work in your final project, final presentation, and in your professional math journal.

Recall also that SU expects all students to spend roughly 2 to 3 hours outside of class for every hour in class, on outside work. For Math 190 this means about 6 hours each week should be spent (probably less during some weeks, and more as you work on drafts for your projects, PEMJ etc to be handed in). Do not let yourself get behind just because not every assignment is collected. Your participation (or lack of participation) in the in-class discussions of previously assignment homework problems (like next week's discussions of the Google Calculator and Modeling and Estimation problems) will count as part of your grade.

The above four bullets are this week's assignments. However, your weekly work should also always include reviewing your classnotes; as the semester progresses, you will also want to start researching online -- some weeks there may be an assignment to try to find, for example, a mathematically misleading headline or line in an article -- and perhaps offline on mathematical subjects and environmental contexts, to enrich your undrestanding, to share examples in class discussions, and eventually to help you select your Project topic. Final note: you are welcome to work in groups of 2 or (at most) 3 students on such homework assignments, so long as your write-ups (whether on PEMJ or other items later due to be handed in) gives credit, by name, when your learning benefitted from work with others, and making sure your own understanding is solid so as to be able to do well on not just the mini-tests but also the written and project assignments.

Homework: Modeling and Estimation

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.

In this exercise you will be asked to do two things: to create a simple mathematical model for a quantity, and to estimate that quantity. To illustrate the difference between the two, here is an example. Suppose the question is:

SU wants to create an "SU goes Tropical" theme weekend in which the main grass quadrangle next to Henson is turned into a "Beach" [Suppose a precise map of picture is given of the rectangular region in question]. How many cubic feet of sand would SU need to use to accomplish this, if the sand is to be a half-foot deep?

This is not the assignment this week! But, if we were asked to give both a model and an estimate in response to this (admittedly a tad environmentally costly) scenario, a solution might look something like this:

"Modeling: The 3-dimensional region that the sand occupies will be geometrically like a rectangular box: it will have a length, a width, and a height. Its height (or depth) in feet will be ½ foot or 0.5 feet. In my model I will use "x" for the length of the rectangular region and "y" for the width of this region. Thus the volume of the amount of sand that would be there, which is the amount of sand SU would need, would be (length)·(width)·(height) which is ½·x·y. Note that here x and y must be measured in feet, and that ½·x·y will give us the answer in cubic feet"
My Estimate: I took the map provided and went outside of Henson. I used [tape measure? Yardsticks? Some other estimate? The student would explain this here] and estimated that x is about 42 feet and y is about 27 feet. Thus the volume of sand needed, per step i. above, would be ½·42·27 cubic feet. This comes out to 567 cubic feet ( 567 ft³).

Notice that two people might use the same model, but have different estimates (in problems that are less simple, like this HW problem, there may even be two or more different, workable models that are possible, before we even get to estimations). Now that you have an understanding of what it means to create a model and to come up with an estimate, create both a model and an estimate for each of the two questions below. For simplicity, assume that each student takes 2 pieces of paper for scratch-work, and assume that none of students recycles any of the scratch-paper.
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.
2. Create a model for, and come up with an estimate for the number of pieces of paper thrown away and not recycled counting (from tests' scrachpaper only; not all sources of paper waste) from all classes at SU (not just Dr. B.'s), per semester. (You may want to google "2008 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)
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. Your task is to gain experience with and familiarity with mathematically modeling real-world situations, and making reasonable estimates based on them. Sample (possible) solutions will be handed out next week. So roll up your sleeves, work in groups if you wish, and have fun with this exercise.