Modeling Oscillating Systems: Mathematics Component
Activities and Procedures
In a whole class setting, the lab reports from the physics class will be reviewed to develop an understanding of the symbolic model parameters that will affect the data for the various experimental situations.
The teacher should discuss how the data these reports are based on was created, emphasizing that a model developed from experiential data created the data.In other words the data is not from an experiment but from a model based on an experiment.
The teacher may want to represent one example set of data using the graphing calculator to reinforce the concept that the data is periodic and that a trigonometric model is appropriate.
Students will have learned how the coefficients of the sine function affect the graph. Introduce at this point the procedures for calculating these coefficients from a graph of the data. Provide additional practice in this skill. Supply various sine graphs and allow students to determine the sine functions.
Place the students in groups of no less than three and no more than five. Divide the data sets among them in an organized manner. Guide them in determining the most efficient distribution of data sets.
It’s important that the students understand the goal here. Not only are they developing a mathematical model based on this data, but they are also looking for a relationship between the symbolic model parameters and the coefficients of the mathematical model.
At the very least, the data should be divided by experimental situation (i.e. spring, pendulum, etc…). In addition, each group should have sets of data where all but one model parameter has been held constant thus allowing the student to determine a relationship between the mathematical model and that specific model parameter.
The logistics of installing the data sets onto the computers needs to be worked out.This should be done before hand, not during class time.
In the computer lab, each student will be working on their own computer, but sitting with their group members.
Students will start Excel and open the prepared notebook (download oscillation.xls). Students will copy and paste their data sets into the notebook and begin analysis. A basic instruction sheet will be provided.
Students will use a structured report format to record the results of the following analysis.
From the graphs that Excel will provide from the copied data, students will calculate the coefficients of a sine function, enter them into the appropriate places in the spreadsheets, and test their model. In other words, students will be finding the sine function whose graph matches the graph of the symbolic model data provided by the physics class. They can then “tweak” the coefficients to determine the mathematical model as accurately as possible.
This process may take a couple of days. There will be many data sets to analyze, determine good mathematical models for, and then make the connections to the symbolic model parameters. The level of student will determine the level of teacher guidance during this activity.
Once they have established mathematical models for their data sets, they will make conclusions about how a specific model parameter is affecting the mathematical model and record this information into a structured report format. Discuss with the class about organizing all the model reports, and grouping them by varying model parameter. For example, they will collect a group of models that come from the spring and mass experiment where different masses have been used, but all other parameters are constant. Then, by looking at how the mathematical model changes along with the change in the symbolic model parameter, a relationship can be determined.
Again, this process of developing models, grouping them appropriately, making conclusions, and writing up reports may take several days of labwork.
A wrap up day with basic modeling practice, and making sure objectives have been met may be a good idea before the final unit assessment.
A performance task and an exam will follow the above activities. Refer to the assessment section for details. The teacher should decide how these assessments are administered. Decisions to be made include working in groups or individually, working inside or out of class, and duration of time over which the assessment will be administered.