## Bouncing Tennis Balls

- Lesson

Students develop their skills in collecting and recording data using the real-world situation of a bouncing tennis ball. They use the data collected to formulate the relationship between the dependent and independent variable in their experiment.

### 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.

(Seconds) | during Interval | of Bounces |

### Graphing the Data

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).

- Bouncing Tennis Balls Recording Sheet
- Tennis balls, one for each team of four students
- A clock or watch with a second hand
- Centimeter graph paper, a spreadsheet program, or a graphing calculator (optional)

**Assessment Option**

During the activity, it is important to observe the students. Note which students made graphs correctly, paying attention to how they used the idea of scale to set up the time and distance axes. Listen to students' conversations about their graphs, attending to comments that indicated the students realized that the number of bounces depends on the length of time the ball is bounced and that patterns develop when the ball is bounced in a consistent way.

**Extensions**

- 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.
- 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?

*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).]

**Questions for Students**

Refer to the lesson plan.

**Teacher Reflection**

None

### Learning Objectives

Students will:

- Collect data and record data in a table.
- Make a graph to display data using correct labels and scale.
- Recognize what varies in an experiment.
- Name the independent and dependent variables in a problem.

### NCTM Standards and Expectations

- Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

### Common Core State Standards – Mathematics

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Grade 6, Stats & Probability

- CCSS.Math.Content.6.SP.A.1

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, ''How old am I?'' is not a statistical question, but ''How old are the students in my school?'' is a statistical question because one anticipates variability in students' ages.