Intent

In this, the unit’s ultimate activity, students look for a function to fit their pendulum data and to enable them to predict the period of a 30-foot pendulum. Then they measure the time needed for a real 30-foot pendulum to make 12 swings and compare their experimental results with their predictions.

Mathematics

Students will begin by trying to fit a curve to the data they collected in An Important Function. They will use the function, f, that they find to compute f(30). They will then test this prediction by building and timing a 30-foot pendulum.

Progression

Students will work on the curve-fitting part of this activity in groups, share their predictions with the class, and then build and test a 30-foot pendulum as a class.

Approximate Time

40 minutes for activity

35 minutes for discussion

Classroom Organization

Groups and whole class

Materials

Class data table from An Important Function

Materials for building a 30-foot pendulum, such as nylon fishing line and a 5-pound weigh (Sufficient weight is needed to keep the line taut.)

Doing the Activity

Students have finished almost all the necessary preparation for answering the unit problem. They have found out which variable affects the period, and they have collected relevant data. They are now ready to look for a pattern in their data and use it to predict the time it will take for a 30-foot pendulum to make 12 swings.

Remind students of the curve-fitting they did using graphing calculators in TheOverland Trail, and explain that they will use the same technique to analyze their pendulum data. You may want to review the general steps.

If groups need a hint as they look for a function, ask whether the graph resembles any of the examples from recent activities, or have them look at the posters from Graphing Free-for-All and Graphing Summary.

Ask students, What function did you find to fit the data? If they have accurate data with a wide enough range, with L representing the pendulum length and P representing the time for 12 periods, they should find that there is an equation of the form P=cLP=cL size 12{P=c sqrt {L} } {}, for some constant c, that closely fits the data. You may get several curves of this form. The class need not agree on which function best fits the data.

What is your prediction for the period of a 30-foot pendulum? Collect and post groups’ values for f(30) for later comparison with the results of the upcoming experiment.

Raise the question of whether extending the data in this way makes sense. Do you think this is a reliable method for making a prediction?Do you have any reason to believe that the pattern of the data will continue as far out as 30 feet?All you need to accomplish here is to raise some skepticism. Students’ curve-fitting should work well in this situation, but they should be aware that this is a complex issue. The real test will come when they actually conduct the 30-foot pendulum experiment.

To construct the pendulum, you might hang it from the back of the bleachers in the school stadium or gymnasium or ask the fire department or a utility company to provide a truck from which the pendulum can be swung.

If you are unable to set up a 30-foot pendulum, build one that is as long as possible and have students predict the time it will take for a pendulum of that length to make 12 swings.

Have several timers time each swing, and average their results.

Discussing and Debriefing the Activity

Once the data have been collected, discuss whether the functions students found accurately predict the period of the actual 30-foot pendulum. If they do not, let students speculate why not.

Then turn back to the opening story and ask,Do you think Poe’s story is realistic?Do you think the prisoner could have escaped in the amount of time it takes a 30-foot pendulum to swing 12 times?

To give students a better sense of how long that time is, ask them to close their eyes and try to estimate when a time interval has elapsed that is equal to the time they measured for the 12 swings of a 30-foot pendulum.

Key Questions

What function did you find to fit the data?

What is your prediction for the period of a 30-foot pendulum?

Do you think this is a reliable method for making a prediction?

Did your predictions match the actual period?

Do you think the prisoner could have escaped in that amount of time?

Supplemental Activity

Out of Action (reinforcement or extension) asks students to fit a curve to data to make a prediction and then use that prediction to make a decision. This activity might be used late in the unit, as a follow-up to students’ work on the unit problem.