How High Will it Bounce?

How high will it bounce?


Lesson Objective:

Students will generate data and model the data collected with tables, equations, and graphs.  They will calculate the rebound ratio when a ball bounces.  The function is linear.

CCS Standard(s):

F‑IF.7e, F‑LE.1c

Mathematical Practices:

make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically

Lesson Mathcast:

 Mathcast 

Lesson Video:

 The Bouncing Ball 

Length of Activity:

One day (approximately 50 minutes)

Core Problems:

Problems 1 through 4

Materials:

Small, rubber, very bouncy balls, one per team

Meter sticks, two per team (or one longer measuring device per team, as explained in “Materials Preparation” below)

Graphing calculator with display capabilities (optional)

Materials Preparation:

To measure the rebound height of a bouncing ball, each team will need a measuring tool set up against a wall. Meter sticks or tape measures work, or you can set up a strip of paper (such as adding machine tape) with one-centimeter increments marked ahead of time.

Alternatively you could post a blank strip of paper (about 10 inches wide and 5 feet long) vertically on the wall for each team. Students mark all the drops from one height with one color pencil, and then change colors for the next height, and so on. Students measure after they are all done bouncing. This set up is easier to execute if you anticipate students having difficulty bouncing a ball straight up along the meter stick.

Technology Notes:

 

You can optionally direct students to create a scatterplot on their graphing calculators. If you are using TI-83+/TI-84+ calculators, find more notes about how to create scatterplots in the Calculator Instructions available at cpm.org. Of particular interest will be these section in the Calculator Instructions:

  • WHOLE-CLASS PROJECTION
  • GETTING STARTED --- ADJUSTING SETTINGS
  • ENTERING DATA
  • BIVARIATE DATA
  • THREE REPRESENTATIONS OF FUNCTIONS


Before class, be sure to set up and test your graphing technology to make sure batteries and software work as planned. It also is a good idea to walk through the activity using the technology yourself on the day of the lesson to review the buttons and issues that may arise during the lesson.

There are a few options for teachers to demonstrate the graphing calculator to the whole class. See the “Whole Class Projection” section.

If you are using Desmos, refer to the tutorials for creating tables and scatterplots. Desmos is a free, online graphing calculator.

Suggested Lesson Activity:

After a brief lesson introduction, give students 5 minutes to respond to problem 1. As students share their ideas, be sure that the idea of using the ratio  to quantify each ball’s “bounciness” comes up.  This could be done as a Think-Ink-Pair-Share.  The Ink is in case you do not need the teams to record their answers.

Move teams on to problem 2.  Give teams a few minutes to discuss the questions in the text and plan their experiment.  This can be done as a Teammates Consult.  Teams should call you over when they think have a viable plan.  A viable plan should include that students have agreed upon what data they are collecting (height dropped from and rebound height) and how they are recording this data.  A viable plan might look like this:

  • We will measure at 120 cm, 100 cm, 75 cm, 65 cm, and 40 cm  (Students should measure several drop heights, and the intervals between drop heights should not be evenly spaced)
  • We will measure 3 times at each height
  • We will use the bottom of the ball as the place where we measure the height
  • We will record our data in this table:

Ask the team to describe their plan and, if you think they are ready, provide them with a ball and measuring device and have them start.  If their plan does not yet seem viable, point out to them the ways that you are not sure it will work and give them more time to revise it and call you back again.  Another option is to use a Fishbowl to demonstrate what the activity will look like.

When teams have collected data, they should move on to problem 3.  As teams work, check their graphs for completeness and accuracy.  Make sure students identify the starting height as the independent variable and the rebound height as the dependent variable.

As they graph their data, they should “eyeball” a line that fits their data.  Since students do not use a starting height of zero, they may not realize that they should include the point (0, 0) on their graph.  Some teams may need reminding that in mathematics, a “line” always refers to a “straight line;” otherwise the word “curve” is used.  Students should have experience with “eyeballing” a line of best fit from previous courses.

Make sure students understand how to interpret the y‑intercept as it relates to their task.  As you circulate, you might ask teams, “What does the y‑intercept mean in this situation?” or“What will the rebound height be if the ball is dropped from a height of zero centimeters?”

Students should continue to work on problem 4.  They should recognize that the slope represents the ball’s rebound ratio.  When problem 4 is finished, you could use a Traveling Salesman strategy to share answers.

Direct students to save their data and their work from today in a safe place.  It will be needed for the next lesson.

If time allows, you may want to show students how to enter data points into graphing calculators, and to make a scatterplot.  This could be a whole‑class activity, or you could use it for teams that finish ahead of the others.  Using a graphing presentation tool, one team could enter data into it and display it for the class.  Students can verify that the scatterplot on their calculator matches the graph they made in part (b) of problem 3.  Show students how to appropriately set the windows of their calculator screen. 

Guide students to enter the equation of the line they found in part (c) of problem 3: to input an equation for graphing, students should press .  Students can now graph their scatterplot with their line overlaid. 

Closure:
(5 minutes)

Ask students to describe what they now know about their rebound ratio.  Take a few minutes for students to help each other clarify any points of confusion from the activity using a Walk and Talk or Proximity Partners.  Tell students that in the next lesson, they will use the information that they found today to make predictions about the behavior of their bouncy balls.

Team Strategies:

If you are using team roles in your class, use the General Team Roles Resource Page before the activity begins to remind students of the words you expect to hear as they perform their roles today.

For problem 1, use a Think-Ink-Pair-Share.  Then ask Facilitators to make sure problem 2 is read out loud and that each team member is taking part in the discussion and formation of the plan.  Facilitators can then make sure that their teams are clear about who will be the ball dropper, data recorder, and spotters.  Resource Managers should be responsible for collecting the ball and any other resources as well as call the teacher over when the whole team has a question.  Task Managers can remind the spotters to measure the rebound height to the same place on the ball for each trial.  Recorder/Reporters should explain the team’s plan for collecting data and make sure that each person records data on his or her own paper.

Universal Access:

 

Students will ask what “Squash” is.  Although not critical to answering the problem, you may need to satisfy their curiosity, or else they will make up what the phrase “Squash balls” refers to.

For teams to complete problem 2, if you are concerned about time, consider preparing a resource page for students to complete that has the outline of a table.  You can decide how much information to start them off with.  Whenever using several different materials and supplies, things always take longer than they should, and getting in appropriate discussions and having time for the students to collaborate within a period as little as 45 minutes can be stressful.  Depending on the strength of your low readers and EL students, preparing an outline ahead of time will be a useful and necessary scaffold for your students.

Perhaps one side of a poster can have the table, notes for them to remember, and directions for handling the activity and materials.  The other side could be a graph already set up.  Although there is value in having the students discuss and create the organizational tools themselves, there are always time limitations and learning considerations as well.

Answers:

1.  Teams should come to the idea of using this ratio: .  The basketball is bounciest with a rebound ratio of 0.743.

3.      a.    Independent: starting height,  Dependent: rebound height.

          b.    The data should be approximately linear.  Yes; the rebound ratio is constant.  The line should pass through the origin, because when the starting height is 0, the rebound height is 0. This is a proportional relationship.

          c.    Answers vary but should be y = (rebound ratio) x where x represents the starting height and y represents the rebound height.

4.  Rebound ratios vary for different balls. The rebound ratio is the slope of the line of best fit. The rebound ratio is the coefficient of x in the equation of the line. In the table, the rebound ratio is .