**Students often have a difficult time interpreting data from a graph of two variables when the relationship between these two variables doesn't exist. Teachers should approach these situations with care, and use them as an opportunity for direct discussion of what "no-relationship" means in a scientific investigation.**

On a recent visit to a New York City public school, I saw ninth graders complete a classic pendulum lab. Students measured the period of a pendulum at various different angles of initial displacement, and then measured the period for various string lengths. Students then graphed their results and posted them on the walls of the room for everyone to see. Though not many groups looked around the room at their classmates' graphs, if they had they may have been surprised at the diversity of results! I've included a sampling of some student graphs showing their results concerning the relationship between period and amplitude.

These graphs illustrate how challenging it is for students to accept the

*no relationship*relationship. (Before you take me to task about the amplitude-dependence of the period of a pendulum, keep in mind that for the maximum amplitude measured here, the actual period deviation from a small-angle approximated pendulum is less than 5%...) Young people who have been graphing results in science class since elementary school are so used to seeing a trend in the relationship between two variables that they go bonkers when the relationship between those variables doesn't jump out at them.
Watching students carry out this pendulum lab was fascinating. Nearly every group doing the lab expressed doubt and dismay when they noticed they were getting approximately the same value of period for every amplitude. (At least one group decided to do something about it: if they didn't get a result that was sufficiently different from the last angle they measured, they went back and did the run again.) One member of the group that made the correct graph shown on the upper right apologized that they "did the experiment wrong." The assumptions students brought with them to this lab led to quite a few discussions with the teacher about why they were so sure that their measurements were wrong, but I didn't get the impression that these conversations resulted in much self-reflection. (We could find out, of course, by asking these same students to examine the relationship between period and mass...)

This period-amplitude relationship was just one piece of the entire pendulum lab, and students' graphs showing the relationship between string length and period were more successful than the period-amplitude graphs I've shown here. But, applications in torture aside, it seems like pendulum's most promising role in a Physics First course is in encouraging students to examine their own role as scientists. With a little bit of restructuring to this activity, the widely varying results shown in these graphs could provide the fuel for quite a sophisticated discussion about experimental design.