### The Activity

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

**How many times can each team member bounce and catch a tennis ball in two minutes? **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.

**A Sample Data Set for Bouncing Tennis Balls**Time (Seconds) | Number of Bounces during Interval | Cumulative Number of Bounces |

0 | 0 | 0 |

10 | 11 | 11 |

20 | 11 | 22 |

30 | 9 | 31 |

40 | 10 | 41 |

50 | 11 | 52 |

60 | 10 | 62 |

70 | 11 | 73 |

80 | 11 | 84 |

90 | 10 | 94 |

100 | 10 | 104 |

110 | 10 | 114 |

120 | 10 | 124 |

### 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.

*A graph made using a graphing calculator*### 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).