The Activity 

Students form teams of four to bounce a tennis ball. Pose the problem problem:

A bounce is defined as dropping the ball from the student's waist. One student keeps the time while the second student bounces and catches the ball, the third student counts the bounces, and the fourth student records the data in a table showing both the number of bounces during each ten-second interval and the cumulative number of bounces. Each trial consists of a two-minute experiment, with the number of bounces recorded after every ten seconds (or twenty seconds for fewer data points). The timekeeper calls out the time at ten-second intervals. When the time is called, the counter calls out the number of bounces that occurred during that ten-second interval. The recorder records this count and keeps track of the cumulative number of bounces.

The same process is followed by each student, with the students rotating roles, so that each student can collect a set of data. All the students must bounce the ball on the same surface (e.g., tile, carpet, concrete) because differences in the surface could affect the number of bounces.

Distribute the Bouncing Tennis Balls Recording Sheet to the students.

  Bouncing Balls Recording Sheet

The data from one student's experiment are recorded in the table below.

Graphing the Data 

 Line of Best Fit Tool

Once the data have been collected, each student prepares a graph showing the cumulative bounces over two minutes. This graph can be constructed by using the Line of Best Fit Tool.

Alternatively, students may graph the data by hand, by using a graphing calculator, or by using a spreadsheet, depending on the students' experiences and on what information the teacher wants to gather about what the students know and are able to do.

The image below shows the data plotted using the Line of Best Fit Tool.

Alternatively, students may use a graphing calculator to display their data. The figure below shows such a display.

Discussion 

Students present their results to classmates by showing their graphs. The discussion can involve what the students found easy and what they found difficult in completing this task. Students' discussions can be revealing. During the discussion, think about these guiding questions:

• Can the students identify what varies in the experiment? Do they comment on the dependent and independent variables either implicitly, in their conversations about the graphs, or explicitly, using correct terminology?
• Do they discuss whether the points should be connected with a line? The numbers of bounces are discrete data, so they should not be connected.
• Decisions about the scale for each of the axes are important. Do the students understand what the graphs would look like if the scales changed?
• When directed to sketch lines on their graphs in order to notice trends, do they demonstrate some sense that the steepness of a line is related to the number of bounces per second?

Your observations related to these and other questions will yield information about what your students appear to know and are able to do that will guide you in making instructional decisions.

Building a Sense of Time and Its Relation to Distance and Speed 

Initially students need to become aware of their own understanding of time, change over time, and the use of new kinds of measure (i.e., rates). Posing such questions as those listed below focuses their attention on these ideas (adapted from Kleiman et al. 1998).

• How do you measure time? Distance? Speed? 
• Give an example of something that might be able to travel at two feet per second. 
• What is the difference between traveling at two feet per second and two feet per minute or two feet per hour? 

In this context, distance is how far the object or person moves (travels). Speed is how fast the object or person is moving (traveling). Both are described in terms of direction. Distance is measured in such units as feet, miles, or kilometers. Speed is measured in relation to time using units such as meters per second or miles per hour.

Reference

Adapted from Friel, Susan, et al, Navigating Through Algebra in Grades 6 - 8, from the Navigations Series, NCTM (2001).

1. An extension of this activity would be for each student to conduct an experiment using, for example, concrete floors and then carpeted floors to investigate the effect of differences in the surfaces.
2. Students can also explore different activities that test their sense of time. They can do the following activities in pairs; in each instance they many want to observe if they overestimate or underestimate the time and try the task again.
   • Clap your hands so you clap exactly one clap per second for ten seconds. 
   • Turn a page in a book at exactly one page every two seconds for twenty seconds. 
   • Sit still for thirty seconds, letting the timer know by raising your hand when you think thirty seconds has passed. 
   • Walk at the speed of one foot per second for fifteen seconds. 
   • Walk the length of your classroom in exactly ten seconds. At what speed were you traveling? 
3. Connecting Graphs to Stories: In "Walking Strides," students examine how the time required to walk a given distance varies as the length of their stride varies. Often it is reported that many students initially misjudged time. A suggestion to overcome this problem is to let them explore the activities in teams: have them sit for an undisclosed time (e.g., 30 seconds) and make guesses about the amount of time that had elapsed.

   [This activity has been adapted from Jones and Day (1998, pp. 18-19).]