UNIT I

• 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.

• 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.

2. Based on your experimental results:

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

1.2 On the Road

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.

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?

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.

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?

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?