Mathematics at the introductory level should be taught as a laboratory science. Stevens (1993) presents several examples of sources of data that lend themselves to laboratory modeling projects. They include the heights of corn seedlings as a function of time, normal high temperature as a function of the day of the year, and the height of a burning candle as a function of the length of time that it has been burning. Students might also attempt to model the price of one brand of laundry detergent as a function of the weight or volume of the detergent, or develop a population model for a country like Mexico over the past 30 years. In some situations, scientific principles or previous experiences indicate the type of model that should be developed. In other cases, students have to experiment with a variety of models to determine the one that fits best.
Once the data have been gathered and graphed, students can determine the parameters of the hypothesized model algebraically, through trial and error (seeing how well the proposed model fits the points), or by using regression programs on a calculator or computer.
The following project does not need any special data gathering equipment other than a stopwatch. It is assumed that students are working in groups.
Introduction. As liquid flows through a hole at the bottom of a container with a constant cross-sectional area, the height of the liquid above the hole is a function of the time that the liquid has been flowing. The fact that the function is a quadratic function is an application of Torricelli's law.
Objective. Students will gather data and fit the data to a quadratic function using educated trial and error. They will determine approximations of the needed parameters algebraically and then refine their approximations graphically.
1. Calibrate the height of a clear plastic container that has a constant cross-sectional area in centimeters. Drill a small hole at zero. it is best to put the zero point a few centimeters up from the bottom of the container. (The whole container does not have to have a constant cross-sectional area. Simply calibrate the portion that is constant.) A height of about 10 cm is sufficient. Fill the container with colored water to the 10-cm mark while holding your finger over the hole.
2. Release your finger from the hole and at the same time have a groupmate start the stopwatch. Determine the length of time that it takes to pass the 8-cm, 6-cm, and 4-cm marks and record the results. You might want to do the experiment three times and use the mean time at each level.
3. You must now determine a quadratic function
that best fits the data. Plot your four data points [(0,10) is the first point]. Use algebra to determine a quadratic function through three of the points. Modify the parameters of that function so that it "best" fits all four of the data points. Here, the best fit is simply determined by looking at the graph.
4. Once you have determined a function that closely fits all four data points, predict when the fluid will be at h = 0. Fill the container to a height of 10 cm and measure the time that it takes the water level to get to zero. How accurate was your prediction?
Conclusion. Write a report that explains the procedures that you used and provide a description of your results. Explain the reasoning that you used. Specifically explain how changing the parameters affected the shape of your graph.
Possible Model. A one gallon window washer fluid container was used to gather the following data.
Figure 4 represents the graph of . Improvements can be made to the fit by changing the coefficients of and t.
Figure 4. The graph of