# UNIT I

## What Is A Mathematical Model?

#### J. Fey Office in 3113 Mathematics Building Phone: x53151 (voice mail) E-Mail: JF7@umail.umd.edu

The word model is used in many different ways in the English language. We speak and hear about fashion models, model airplanes, model students, architectural models, and so on.

What do the many different uses of the word "model" have in common?

What do you imagine the phrase "mathematical model" might mean?

The goal of this first short unit of MATH 110 is to give three illustrations of mathematical modeling--to set the stage for more in-depth investigations of important modeling concepts and techniques in the balance of the course.

In mathematics the word model is both a noun and a verb, an object and a process of thinking. The three modeling activities in this introductory "sampler" unit are titled:

Living Dangerously

In each case you will be asked to explore several practical questions with basic techniques of mathematical modeling. The goal is to gain familiarity with some of the types of mathematical models that are commonly used, the processes involved in building and using mathematical models, the basic tools for analyzing and reporting data from mathematical modeling investigations, and strategies for working collaboratively with other students in mathematical problem solving.

### 1.1 Living Dangerously

Have you ever walked across a very shaky bridge, wondering if it would hold your weight or break? Do you get a bit nervous in a car or bus or train when crossing a high bridge? Over the past several years dramatic accidents have involved collapsing bridges and parts of buildings. For example, the mezzanine in a Kansas City Hotel collapsed when a crowd of party-goers got too lively dancing in rhythm; a bridge on Interstate 95 in Connecticut collapsed unexpectedly dropping several cars into a deep ravine and closing that busy highway for months. That's to say nothing of the building and bridge collapses connected to earthquakes and hurricanes! We certainly all hope that engineers are sound principles and data about strength of materials and structures.

Breaking Bridges -- Most bridges are built on a frame of steel beams--usually covered with concrete also. Steel is very strong, but if you put enough weight on any beam, it will bend or break. The amount of weight that can be supported is related to the length and thickness and design of the beam.

To design a bridge with steel beams it is important to know those relations accurately. When engineers design a bridge they consider strength of the building materials and the support that the bridge design will give to those materials. They use information from tests on scale models.

You can do some experiments just like those that engineers would do to test strength of bridge beams and discover some of the principles involved. You can do it with a ruler, scissors, paper, and pennies as test weights.

• Cut strips of copier paper in various lengths (30, 24, 18, 12, 6 cm), each strip about 5 cm wide. Then fold the edges up about 1 cm on the two long sides.

• Place the "bridge" between two books, with about 2 cm overlap on each end, and stack pennies at the center of the bridge until it crumples.

1. Before you test paper bridges of various lengths, make some estimates of the relationship you expect to find between bridge length L and breaking weight W?

2. Test your ideas about the relation between bridge length and strength by finding the breaking weights for paper bridges of length 30, 24, 18, 12, and 6 cm. Describe the pattern in that data and compare it to your prediction in (1).

3. Use the pattern in (2) to make (and then test) breaking weights for some other intermediate lengths.

4. Write a report describing your findings as clearly and accurately as possible.

5. As an extension, make some conjectures about how breaking weight would change as bridges of different width or thickness are tested. Design and conduct strength tests of thicker or wider paper bridges to confirm your ideas.

Conclusions and Connections -- In the bridge experiment the quantities of interest, the weights and the lengths of bridges, are variables. In the experiment you manipulated one variable and observed the response of the other. The goal was to find mathematical models of the relation between the manipulated and responding variables. Those relations could be expressed in a variety of ways:

• In English language sentences;
• In tables of (bridge length, breaking weight) data;
• In plots of the data pairs;
• In graphs of the relations between the variables;
• In symbolic equations.
1. What are the manipulated and responding variables the bridge breaking experiment?

2. Based on your experimental results:

a. Sketch a graph that expresses the relation among variables you discovered.
b. If possible, find an equation that relates the variables in the pattern of that graph.

3. Describe some other familiar situations involving related variable quantities and identify the manipulated and responding variables.

The diagram on the next page is a kind of map--showing distances in miles between major U. S. cities. The map shown is one example of a mathematical model called a network or vertex-edge graph. Use the information in that model to answer the following questions that are typical of those in graph model problems:

1. Pick two cities--one in the northeast section of the map and another on the west coast--and find the shortest path from one to the other. Be prepared to explain how you know that you have the desired shortest path.

2. The cities on the map are all connected by interstate highways. It's probably fairly clear that some of the highway links could be removed without isolating any cities. What set of interstate highway links would give the shortest total system on which it is still possible to reach each of the cities shown? Again, be prepared to explain your answer.

3. The following sketch could be that of a much simpler road network connecting only 7 cities. Try again to find the set of highway links that would give the shortest total system on which it is still possible to reach each of the cities shown. Again, be prepared to explain your answer.

4. Look back at the problems (1) - (3) and ask yourself this question: "How could a machine (like a computer) be given directions to solve the problems." That is, ask yourself whether your methods for answering the various questions could be described as a clear procedure with simple steps that could be executed by a machine.

Use this page for map; blank in regular file to preserve page

number sequence.

Conclusions and Connections -- The kind of mathematical network model presented for interstate highways here is used in a variety of situations by airline route planners, telephone and cable tv line installers, trash hauling companies, security guard services, and many other businesses.

1. Think of problems that might be faced in those industries that would involve essentially the same mathematical issues as the three questions posed about highway networks.

2. The interstate highway network and the simpler "road" network diagram are not drawn to exact scale, but significant information is indicated on the diagrams. In what sense does the network diagram still act as a model of the real highway system?

3. As you worked on the interstate road network questions, in what ways did you apply information about U. S. geography not actually given on the diagram--information and insights that would be hard to communicate to a computer procedure for finding shortest paths and networks?

In a mathematical model for bridge building we want to know quite precisely how the modeled system will behave. For example, we want to know just how much weight a bridge of certain length and thickness will support safely. In many important problem situations such precise predictions are not possible--there is an element of random variation.

Think about the business of selling shoes in a busy mall store. Surveys show that on a typical day several hundred customers visit the Foot Locker store in a major shopping mall, but only about 2 out of 5 customers actually buy something there.

What factors will affect sales at such a shoe store on a given day?

Suppose you worked in that Foot Locker store and in one hour only 4 of the 15 customers you waited on actually bought shoes.

1. Would it be fair for the manager to judge you as an ineffective salesperson?

2. If another salesperson actually sold to 8 of 15 customers, would it be fair to judge that salesperson as especially effective?

There is a simple way to study the questions posed in (1) and (2)--to see whether the differences in sales results are due to skill of the salesperson or just the result of random variation in customers who happened to walk in the store. The next page shows a small table of random digits. It was formed by picking digits 0, 1, 2, ... , 9 one at a time as in a lottery drawing where the balls are replaced after each selection.

• Think of each digit as a Foot Locker customer--digits 0, 1, 2, 3 represent buyers and digits 4, 5, 6, 7, 8, 9 represent non-buyers.
• Break the list into groups of 15 and count the number of "buyers" in each of at least 25 such groups.

3. Use the data from your simulation of shoppers to prepare a report to the store manager addressing the following questions:

1. In a group of 15 customers, what seems to be the most typical number of buyers that could be expected?
2. Is it unusual for a salesperson to get as few as 4 sales from 15 customers?
3. Is it unusual for a salesperson to get as many as 8 sales from 15 customers?
4. What is the longest run of "sales" that might occur purely by chance?

720 Random Digits

01159 63267 10632 48391 31751 57260 68980 05339 26665 55823 47641 86225

31704 88492 99382 14454 04504 66821 41575 49767 04037 30934 47744 07481

83828 59404 72059 43947 51680 43852 59693 78212 16993 35902 91386 48509

23929 27482 45476 04515 25624 95096 67946 16930 33361 15470 48355 88651

22596 83761 60873 43253 8414 20368 07126 20094 98977 74843 93413 14387

06345 80854 09279 41196 37480 61199 67940 55121 29821 59076 07936 11087

96294 14013 31792 87315 56303 08337 52701 93779 76355 98936 00911 90872

18627 00441 58997 14060 40619 29549 69616 57275 36898 81304 48585 32624

68691 14845 46672 61958 77100 20857 73156 70284 24326 65961 73488 41839

55382 17267 70943 15633 84924 90415 93614 99782 93478 53152 67433 35663

52972 38688 32486 45134 63545 59820 96163 78851 16499 87064 13075 73035

41207 74699 09310 56699 31048 43905 47431 72681 55387 77463 26313 91035

Conclusions and Connections -- The table of random digits has been used here as a simulation model to explore questions about random variation in situations. There are many other ways to build simulation models and many other purposes to which such models can be applied.

1. How would you vary the random digit shoe shopper simulation for another store where the typical fraction of buyers was 3 out of 5?

2. How would you design a simulation of shoe shopping if you didn't have a table of random digits, but you did have:

1. A collection of red and black checkers?
2. A standard deck of playing cards?
3. A typical spinner from a board game like that pictured at the right?

3. How could you design a simulation to answer the following kind of question:

In 1994 the Baltimore Orioles won about 3 out of every 5 games they played. However, in one stretch they lost 4 in a row. Were they in a slump then, or was that something that could be expected to occur?

4. Jury selection in major trials is a very controversial process. Both prosecution and defense want juries that will favor their cases. Suppose that in a case of spouse abuse the jury that was finally selected had only 4 women and 8 men.

1. Does it seem to you that this jury was chosen without gender bias?
2. How could you do some experiments to test that likelihood of getting a jury with so few women if jurors were selected at random?

5. In what ways do the random simulation models seem similar to and different from the models of bridge breaking and road networks?

Looking Back

Thinking back over the three sample modeling investigations you've just
completed, how would you now answer these questions:

1. How is the use of model in mathematics similar to and different from the use of that word in
other situations?

2. In what ways can you see mathematical models as helpful tools in solving practical
problems?

3. What do you see as important limitations of mathematical models?

4. How would you explain to someone outside this course what a mathematical model is?

Think about these questions, discuss them with your group partners and the rest of the class,