Can You Carry a Tune in a Bucket?

The ability to generate tunes by blowing air over open or closed pipes is associated with most physics classes. In this variation on a theme, students are asked to generate a tune (up to 8 different pitches, or frequencies) using water-filled test tubes of differing lengths and diameters. The tune must be recognizable by the teacher, and must contain at least 12 separate notes. Suggestions include "Happy Birthday," "Do, A Deer" from The Sound of Music, the Darth Vader theme from Star Wars, and other similar recognizable tunes.

Once the tune has been played, students are given the opportunity to consider the mathematics behind the wavelengths associated with the frequencies generated. Measurements of diameter, the height of the water column in the test tube, and the length of the air column in the test tube are among the measurable parameters students are asked to find. These values are correlated with the wavelengths, frequency or pitches associated with those measurable parameters.

Using the measurable parameters as the independent variables, the wavelength or frequency is treated as the dependent value. Students are asked to use graphing calculator or computer technology, generating graphs that will help explore the mathematical model associated with the data collected from the experiment. Models to be explored would include linear regression, or higher level regression models. (found on most TI Calculators under the Calculate menu associated with the STAT functions, or found in computer software packages like Vernier's Graphical Analysis). Students are encouraged to generate at least three potential models, with corresponding graphs, and provide a reasonable analysis of the Regression Coefficients to determine which model best represents the data, providing reasonable justification for their conclusions.

This lesson, while seemingly simple, generates some wonderful and reproducible data, and has the ability to stimulate student conversation towards mathematical modeling used so much in science disciplines. I have found this lesson to generate great enthusiasm among my students, endears those who are musically inclined to their peers (providing great teamwork and cooperative learning), and gives me a chance to talk with the mathematics teachers about modeling data using linear, quadratic, and higher order regression models. Finally, this lesson gives students a chance to evaluate those mathematical models on the basis of the respective Coefficents of Regression generated, and helps me seque into the topics of interpolation and extrapolation as students use the models generated to predict frequencies or wavelengths not directly measured initially.

This lesson is powerful in the learning opportunities it opens up, and helps the science teacher and the science student integrate mathematical models into the science content.