# Trashketball: Part 1

## Scatter Plots and Lines of Best Fit

K20 Center, Kate Raymond, Levi Patrick, Samantha Marshall | Published: November 17th, 2022 by K20 Center

• Subject Mathematics
• Course Algebra 1, Middle School Mathematics
• Time Frame 200 minutes
• Duration 3-5 class periods

### Summary

In this lesson, students toss balls of trash, or "trashballs," into the trashcan, recording their shooting percentage at various distances from the basket. With distances measured and percentages figured, students create a scatter plot and a line of best fit to make a linear model of the shooting skills of the class. This is a great, simple, and inexpensive lesson that addresses many significant algebra 1 concepts, including expressions, data analysis, scatter plots, and linear equations. Students should understand how to write the equation of a line, given its graph, before participating in this lesson.

### Essential Question(s)

How can linear relations and data analysis help us to understand the abilities of people?

### Snapshot

Engage

Students analyze data that can be used to maximize scores in bowling.

Explore

Students create an algebraic expression for shooting percentages and make a prediction about shooting percentage. They use their expression to verify their prediction and make more predictions about the relationship between the distance from the basket and scoring percentage. Finally, students conduct an experiment and gather data to test their hypothesis.

Explain

Students use their formula and the data they collected to create a scatter plot for the relationship between shooting percentage and distance from the basket.

Extend

Students work together to define correlation, positive correlation, negative correlation, and relatively no correlation. Students use these terms to describe the relationship illustrated by their scatter plots. They then work to create a definition of the term "line of best fit," draw a hypothesis for the line of best fit for the data, and write the equation for their line of best fit. Students then use technology to find the line of best fit.

Evaluate

Students compare and contrast the line of best fit they found by hand with the line of best fit they found using technology. They then use the line of best fit to make predictions about shooting percentages.

### Materials

• Blank paper (one for each pair of students)

• Green sticky notes (one per student)

• Red sticky notes (one per student)

• Tape measures (at least one for every three students; ideally one per student)

• Graphing calculators or spreadsheet software

• Bowling With Jacob handout (attached)

• Trashketball handout (attached)

### Engage

Divide students into groups of two or three. Pass out the attached Bowling With Jacob handout. Have students work in their groups to complete the handout. Each student should record their responses on the handout.

Ask each group to discuss their response to the first question. After all groups have given their answers and reasoning, ask students if they noted any differences in the responses. Discuss any differences noted.

Repeat the above process for the other two questions on the handout.

### Explore

Inform students they will be investigating data about shooting percentages. Ask students to work with a partner to create a definition for shooting percentage.

After a few minutes, ask for volunteers to share their definitions. Record the given definition on the board, and ask other pairs how it differs from their own. Ask if any pair thinks the definition needs to be altered. Continue the discussion until the class agrees on the definition.

Have students work with their partners to create a mathematical definition, or formula, for shooting percentage.

Have each of the student pairs write their formulas on a standard size piece of paper, large enough for the class to see. Post these papers around the room (or have students hold them up) so the other groups can view them. Ask if all of the formulas are equivalent or if there are differences. Have students discuss the differences they see. Ask students if these differences would result in different shooting percentage values and have them explain their thinking. Through this discussion, students should work together to create one formula for the class to use to determine shooting percentages.

Make sure students save the papers on which they wrote their formulas. They will use these papers later to play trashketball.

Give each student a green sticky note and a red sticky note. For each of the examples below, have students hold up a green sticky if they think the example should result in a high shooting percentage, or a red sticky if they think the example should result in a low shooting percentage.

Example 1: 7 attempts, 3 shots made

Example 2: 9 attempts, 8 shots made

Example 3: 5 attempts, 0 shots made

Example 4: 12 attempts, 12 shots made

Example 5: 15 attempts, 16 shots made

Once they have made a prediction for each example, have students use their formula to calculate the shooting percentage. Have students verify that the values they get from the formula match the predictions they made.

Once the class is convinced that the formula they created accurately measures shooting percentage, introduce students to the game of trashketball. Place an empty wastebasket somewhere in the room so that there is space free of obstacles in front of it. The wastebasket can be placed on a desk, table, or on the floor. Wad up a piece of paper and demonstrate shooting it into the wastebasket and retrieving it afterwards.

Ask students to consider if and how moving away from the basket will affect the shooting percentage. Tell students to use the words increase, decrease, or constant to create a hypothesis that describes what will happen to the shooting percentage as the distance between the shooter and the basket increases. Students should record their hypotheses as a complete sentence on the attached Trashketball handout.

Ask students to identify the independent and dependent variables for their hypotheses and record them on their handouts for question 2.

Ask students to identify the control variables and record these for question 2.

Move students, an empty wastebasket, and several tape measures outside (or to the gym, cafeteria, or other area of the school with a large floor space). Have students bring their Trashketball handouts, a writing utensil, and the papers on which they wrote their shooting percentage formulas.

Place the wastebasket in the middle of the floor and use the tape measure to measure a radius of 8 feet from the basket. Have students form a circle around the basket at that distance, standing next to the partner with whom they originally wrote their shooting percentage formulas.

Tell students to tear their formula papers in two. Each partner in a pair gets half of their paper to make a trashketball. Explain that every student will crumple up the half-sheet of paper and try to shoot it into the wastebasket. Ask students what that means about the values in the chart for question 3 of their Trashketball handout. Students should be able to explain that the attempts made will equal the number of students in the class. Have them record this information on their charts and then ask students to make their first attempt from 8 feet.

Once all students have taken their shots, go to the wastebasket and count out how many wads of paper made it into the wastebasket. Have students retrieve a wad of paper (it does not have to be the one they threw) and then form a new circle nine feet from the basket. Continue shooting, recording, and forming circles at distances increasing by 1 foot each time until you run out of room in your space or on the chart.

Once all of the data is collected, let students work in groups of two or three to find the shooting percentages for question 3 of the handout. Students may use calculators but should be sure to check their group's work before recording it on their own papers.

Once all of the shooting percentages are calculated, call on one group at a time to share their result for the shooting percentage at a specific distance. Have the other groups check the other groups' work. All groups should have the same result. If not, have disagreeing groups share their work and reasoning to settle differences. Continue calling on groups to share results until the class agrees on all of the shooting percentages. Once the class has reached agreement, have each student use the data to create a scatter plot for question 4 on the Trashketball handout.

### Explain

Once all students have created their scatter plots, use the I Notice, I Wonder strategy to elicit their thinking.

Explain to students that what they have noticed about the data is that there is a correlation, or relationship, between the variables. Ask students to work in pairs or groups of three to create a definition of correlation.

After every group has created a definition, have one member from each group share their group's definition.

After all groups have shared, ask students what most (or all) definitions had in common, what seems necessary to include in the definition, and what is unnecessary. Create one class definition based on this discussion.

Read the formal definition of correlation given in the "Vocabulary for Teachers" attachment for this lesson. Ask students to compare and contrast their definitions with the formal definition. Allow students to edit the class definition as needed. Once everyone is satisfied with the class definition, have students record it on their Trashketball handouts.

Ask students to return to their groups and consider the next three terms on the handout. Have each group create their own definitions for each term and repeat the procedure above to create class definitions of these three terms. Have students record these definitions on their handouts.

Have students work in pairs to answer questions 5 and 6 on their handouts.

Ask students to raise their hands if they found relatively no correlation in the data. If any groups raise their hand, ask one member of those groups to explain their group's reasoning to the class. Allow other students to ask that student questions or to analyze the group's reasoning.

Next, ask student to raise their hands if they found that there was a positive correlation in the data. If any groups raise their hand, ask one member of those groups to explain their group's reasoning to the class. Allow other students to ask that student questions or to analyze the group's reasoning.

Finally, ask students to raise their hands if they found that there was a negative correlation in the data. If any groups raise their hand, ask one member of those groups to explain their group's reasoning to the class. Allow other students to ask that student questions or to analyze the group's reasoning.

After all three views have been presented and analyzed, poll the class to determine if everyone agrees on the kind of correlation the data represents. If not, continue the discussion until a consensus is reached.

### Extend

Point out to students that their data is close to being linear, but it is not perfectly linear. Tell students that, in cases like this, they will need to work with a line of best fit. Ask students to work in pairs or groups of three to consider the term "line of best fit." What could the definition be? Why would a line of best fit be useful? What would some of the characteristics of a line of best fit be?

After several minutes, each group should share their definitions of a line of best fit. Discuss each definition and what they have in common. Use this discussion to create a class definition of a line of best fit. Have students record this definition on question 7 of the handout.

Next, have each group share one characteristic of a line of best fit. Record the characteristics somewhere all students can see.

Ask students to pick the 3 most important characteristics listed on the board. Discuss these characteristics until the class comes to an agreement about which three are most important. Have students record these characteristics on question 8 of the handout.

Then, direct students to complete question 9 on the handout by drawing a line that fits as many of the listed characteristics as possible. Finally, have students complete question 10 on the handout by writing the equation of the line they drew.

In pairs or groups of three, have students meet in groups to discuss their lines of best fit and complete question 11 on their handouts.

Discuss with the class the similarities and differences they found. Ask students if they can be sure of which line best fits the data. Students should come to realize that, although their equations are all similar, they do not yet have the ability to choose a "best" line.

Explain to students that computer programs and calculators have the ability to find a "best" fit by examining every possible line and finding the one line that minimizes the distance between it and the points on the scatterplot. Demonstrate how to find a line of best fit (linear regression) using a graphing calculator or spreadsheet software and the data from the Bowling With Jacob handout.

After demonstrating using the data from the Bowling With Jacob handout, have students find the line of best fit for the trashketball data.

### Evaluate

Call on one student to share the equation they found for the line of best fit using technology. Have other students verify that they found the same equation and the same line of best fit. If there is disagreement, have students show how they got a different equation. This may involve having students read off the coordinates they used as input, as this is usually the reason for discrepancies.

Have all students use a red pen or marker to graph the new line of best fit found using technology on their scatter plots. Ask students how graphs of the two lines of best fit compare. Which one do they think is a better line of best fit and why?

In groups of two or three, have students complete questions 12 through 14 on their handouts. After all groups complete these questions, have groups share their responses and discuss them with the class.

Allow the groups to complete question 15. After all groups complete these questions, have groups share their responses and discuss them with the class.