**Modeling Patterns in Experimental Data**

** **Many of the most important algebraic forms arise as models of
relationships among scientific variables. Of course, by the time those
relations make their way to the mathematics classroom, they are usually
well-defined and relatively simple expressions. The mathematical modeling
experience in real life is a more complex process--identifying variables,
running experiments that yield data, and finding algebraic expressions that
summarize patterns in the data. The following examples illustrate ways that
such a process can occur.

In preparation for question 2 on the MATH 110 final examination, join with your group partners to select and run one of the following experiments that lead eventually to algebraic models of quantitative relationships.

1.** **Begin this experiment by tossing about 10 coins onto a flat surface.
Count the number of coins that land with heads down and add that number of
coins to the test supply. Note the number of coins added and the new total of
coins in your test supply. Repeat the toss-count-add process to a total of at
least ten trials.

(a) Record the data in a table of data like this:

Toss number 0 1 2 3 4 5 6 7 Coins added 0 Total Number of 10 ... Coins

(b) Plot the *(toss number, total number of coins) * data.

(c) Describe and try to explain the pattern observed.

2. One of the most important scientific principles is illustrated in the typical playground teeter-totter or see-saw. It also comes into play when a diver bounces on the end of a diving board.

(a) What sort of relation would you expect between the length of a diving board
and the weight that it could support before breaking? Sketch a graph of your
conjecture about the relation between *length* and *breaking
weight*..

(b) Using some pieces of fettuccini, a cup hung at the end of the fettuccini diving board, and some pennies, collect data relating length to breaking weight:

Length of "board" cm 24 20 16 12 8 4 Breaking weight pennies

(c) Find an equation that models well the relation between length and breaking weight.