# Bouncing ball experiment

In this experiment students should be in groups of 3. Students will drop a ball from different heights and measure the corresponding bounce. Since each group will use a different ball, they will generate different sets of data. They will be asked to discuss and compare their linear function with that of their classmates. They should practice measuring the ball bounce before they begin to collect data.

**A lesson plan for grades 8–12 Mathematics**

## Learning outcomes

Students will:

- create a scatterplot, both manually and with a calculator, from personally collected data.
- determine the equation of the best fit line, both manually and with a calculator.

## Teacher planning

### Time required for lesson

1 hour

### Materials/resources

- 6-8 bouncing balls
- graph paper
- yardsticks
- rulers

### Technology resources

Graphing calculators

## Pre-activities

- Students must know how to enter ordered pairs on the calculator and how to find the linear regression information.
- They must have experience writing the equation of a line given the slope and y-intercept, or given two points.

Pre-discussion:

- Ask students what will happen to the bounce of the ball when we drop the ball from higher and higher points. Point out that this is an example of direct variation, or that this type of variation is called “direct” because what happens to one variable (increase or decrease) also happens to the other one.
- Could the students predict exactly how high the ball would bounce when dropped from 36″? 72″? How could you make your predictions more accurate?

## Activities

- Practice dropping the ball and measuring its bounce from the first height.
- Drop the ball three times from each height (12″, 24″, 36″, 48″, 60″) before recording the consistent bounce at each height.

**Manually**

- Using your data, plot the ordered pairs on a separate piece of graph paper. Let X be the independent variable and represent the height from which the ball is dropped. Let Y be the dependent variable and represent the height of the bounce.
- Draw the best fit line.
- Write the equation of the line, using a point on the line and an estimated slope, or using two points on the line.

**Calculator**

- Generate a scatterplot using the data from above.
- Determine the linear regression information and correlation coefficient using the calculator’s linear regression capability.
- Write the equation of the line from the linear regression information.
- Graph this equation, along with the one from step 5, on your calculator.

## Assessment

- Using your equation from step 5, predict the height of the bounce from a 72″ drop.
- Using your equation from step 8, predict the height of the bounce from a 72″ drop.
- Actually drop the ball from 72″ and compare the results to your predictions. Which is more accurate? Why do you think this is true?
- Predict how the linear equations of other groups will compare to yours. Are their lines going to have larger slope (steeper), smaller slope (flatter), or the same slope? Why do you think this?
- Write a concluding statement addressing how the correlation coefficient is useful in this experiment.

## Supplemental information

### Comments

The students really enjoy the opportunity to collect their very own data. It is a fun way to practice many of the concepts taught with respect to linear functions and their equations.

- Common Core State Standards
- Mathematics (2010)
#### Grade 8

- Statistics & Probability
- 8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
- 8.SP.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the...
- 8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional...

- Statistics & Probability
#### High School: Algebra

- Creating Equations
- ALG.CE.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

- Creating Equations
#### High School: Statistics & Probability

- Interpreting Categorical & Quantitative Data
- SP.ICQ.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function...

- Interpreting Categorical & Quantitative Data

## North Carolina curriculum alignment

### Mathematics (2004)

#### Grade 9–12 — Algebra 1

**Goal 1**: Number and Operations - The learner will perform operations with numbers and expressions to solve problems.**Objective 1.03**: Model and solve problems using direct variation

**Goal 4**: Algebra - The learner will use relations and functions to solve problems.**Objective 4.01**: Use linear functions or inequalities to model and solve problems; justify results.- Solve using tables, graphs, and algebraic properties.
- Interpret constants and coefficients in the context of the problem.

#### Grade 9–12 — Integrated Mathematics 1

**Goal 4**: Algebra - The learner will use relations and functions to solve problems.**Objective 4.01**: Use linear functions or inequalities to model and solve problems; justify results.- Solve using tables, graphs, and algebraic properties.
- Interpret the constants and coefficients in the context of the problem.

#### Grade 9–12 — Introductory Mathematics

**Goal 1**: Number and Operations - The learner will understand and compute with real numbers.**Objective 1.02**: Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.

**Goal 4**: Algebra - The learner will understand and use linear relations and functions.**Objective 4.03**: Solve problems using linear equations and inequalities; justify symbolically and graphically.