# Middles, Means, and In-Betweens: The Paper Clip Game

Susan N. Friel
Jan Mokros
Susan Jo Russell

Educators and mathematicians are stressing the importance of incorporating data analysis and statistics into the elementary mathematics curriculum to prepare students for living and working in a world filled with information bases on data. In data analysis, students use numbers to describe, compare, predict, and make decisions.

The Paper Clip Game is an activity for fifth and sixth graders who have already had experiences with collecting, organizing, and interpreting data. As they compare two sets of data, they develop ideas about how to choose a few key numbers to summarize data. In this activity, addressed to teachers, we encourage active reasoning about the data rather than reliance on memorized algorithms, such as the procedure for finding the mean. Although students often know how to "add-em-all-up-and-divide," they usually understand little about how the mean represents the data. The complex ideas involved in understanding the mean are developed later in the curriculum module from which this activity is excerpted.

## OVERVIEW

### What Happens

Students play a game in which they blow a paper clip across a flat surface to see how far they can make it travel. They work in teams, play the game, record their results, and display and discuss the class data. Using the same game, they gather data from a smaller, "challenger" group--teachers, students from another class or grade--and again record and display these data. Students compare their class data to the challenger's smaller set of data to answer the question, Which of these groups can make the paper clip travel the farthest?

These activities take two class sessions of about 45 minutes each.

### What to Plan Ahead of Time

• Provide yardsticks or measuring tapes, and paperclips for the game.
• Students will need time for collecting data from the challengers between Sessions 1 and 2.

## Important Mathemtical Ideas

Defining the way the data will be collected. Students make decisions about how to collect the data, play the game, and judge the results. They determine how to define the rules of the game, as well as how to measure the results in a consistent manner.

Summarizing what is "typical" for an individual. Students continue to develop the idea that the accuracy of group data is dependent on having representative and accurate data form each individual. They encounter variability in their own data: "That time I didn't take a deep breath"; "She made me laugh just as I blew the paper clip"; "This was my first try." They consider what a typical distance would be for each student if several trials are permitted.

Comparing Samples that are not equal in size. Students often believe that they cannot compare their data with those from another bigger or smaller group. In fact, having an idea of "what's typical" is very useful in comparing two different-sized groups. With different-sized groups, it is not possible to simply compare totals. Students invent fair ways of comparing the groups.

## SESSION 1 ACTIVITIES

### Introducing the Game

The following introduction is intended as a guideline only. Teachers should modify it to reflect their own styles and the needs of their students.

Teacher: Sometimes, you have to figure out the winner of a game when the sides aren't even. This can be tricky. What do you do when you have to count everybody's score--but the teams aren't equal in size? This is the kind of problem we're going to deal with in this investigation. We're going to devise a game that we'll call "The Paper Clip Game" and compare our scores with those of a challenging team.

The teacher than describes the game, using the following information:

To play this game, students each blow a paper clip across a flat surface like a table or the floor. One way to do this is to make a track out of two yardsticks, placed about two inches apart and parallel to each other. Players are positioned at one end of the track; they blow the paper clip from this end, then measure the distance traveled by checking the yardsticks. A few students may be able to blow the clip farther than a yard, in which case one yardstick can be moved for measuring purposes.

Students may want to experiment with different kinds of tracks for the paper clips, or they many not want to use a track at all. They may want to try a couple of different options, then choose the version of the game that seems to work best for the class.

A variation of this game is "The Pencil Blowing Game." Pencils work better than paper clips on carpeted floors. In one version of this game that we observed, the students extended a measuring tape across the carpeted floor, then stood at one end and blew pencils along a path parallel to the tape. When making their measurements, they decided to count the end of the pencil that was closer to the tape.

The more that students are involved in decisions about the structure and rules for the game, the more interested they'll be in analyzing the outcomes.

Teacher: I have described the point of the game to you; now you need to develop procedures and rules to play the game. How do you want to play it?

Have a few students demonstrate the activity in order to surface questions and develop the procedures they will use to play the game.

Questions that may emerge include:

• How should each of us blow the paper clip?
• Do we blow it from above or from the same level?
• How many tries should people get?
• How should we measure the distance the paper clip travels, especially if it travels at an angle?

As the group makes decisions about procedures, write them down on the board so that players can refer to them during the game.

An important part of this discussion is figuring out which trials "count." Everyone will have a chance to play several times. Do students want to keep only the best result? The most typical result? The best three? They should make this decision as a class and all follow the same procedures.

 DIALOGUE BOX Winners and Losers These students are discussing their line plots (see below) showing the scoring results when their class and a group of teachers as challengers played "The Paper Clip Game." Marta: We decided that our class won. We've got a lot more data in the 26-40 range. Plus our range is bigger. They don't have anyone higher than 32. Lauren: Also, they have two low scores that are less than 20. That's 20% of their scores. We have three low scores out of 19. That's less than 20%. So we win on the percentage of low scores. And we win on the percentage of high scores too. Because we have two high scores and they have only one. Teacher: So comparing the highs and lows made you think that you were the winners. Marta: Yeah, but it's actually pretty close. Teacher: Any other groups? Jacob: First we looked at the medians. Ours is 23 and theirs is 23.5. But that's close to call. Ashok: We looked at the modes. Ours is 24. Theirs is only 23. We win on that one. Lauren: But we have a model at 20 too! Ashok: But we decided to count the highest mode. Jacob: Then we subtracted their high score from our high score. That's a difference of 8 in our favor. Then we subtracted the low scores. That's a difference of 2 in their favor. But overall, we still have 6 extra points, when you subtract it out. Teacher: So your group also compared the highs and the lows, but a little bit differently than the other group. Anyone else? Sirrah: I hate to say it, but we did it a different way and we think the teachers won. (Gasps from the class) Teacher: How did you look at it? Sirrah: Well, if you look a the line plots we made, you can just see that theirs is sort of clumped in the 23-29 area. Ours is more spread out, and we have a lot more people in the 20-22 area. Erica: We think that their typical players are better than our typical players. Julio: Yeah, but the highs should count for something. Our best are better than their best! Teacher: I thin maybe the teachers would be interest in this data. Let's give it to them and see who they think won. Lauren: But they have to be fair about it. They have to give us reasons!

### Collecting the Data: Measuring the Distance Traveled

Students from teams of four or five members to play the game and collect the data for the class. While the teams aren't competing directly against one another, teamwork makes the activity move more quickly. Have students set up a recording sheet to keep track of how far the clip travels each time.

### Representing, Organizing, and Describing the Data

After the teams have finished, students display their results on a class line plot (see example above), perhaps using stick-on notes and a large sheet of chart paper. If students have decided to count a number of trials from each individual, the students might make a separate chart for each trial. Or they can make line plots such as "Our Best Tries" and "Our Typical Scores."

When the class plot is complete, examine the results. Encourage the students to look at clusters of data, gaps in the data, and unusual scores, and to try to describe the overall results.

Keep the class line plot (or other representation of students' data) for Session 2.

### Collecting More Data: The Challenge

With the students, identify a second group of people or "challengers" who will play the game following the same procedures that the students have established. This group might be other children in the school, teachers, or administrators. What's important is that the second group be smaller than your class.

Arrange to have the group come into your classroom to play the game. Be sure the class decides in advance the answers to these questions:

• How many times should the challengers get to practice?
• Which trial or data will we "count" for them?

The challengers could play the game during recess or after school. Students may be willing to let the other group use the honor system and score their own trials. They may find it acceptable for you to monitor the game. However, don't be surprised if a student monitor wants to be present! Use additional record sheets to keep track of the scores of the other group.

## SESSION 2 ACTIVITIES

### Representing, Organizing, and Describing the New Data

Students work in teams, with each team having a copy of the record sheets from the challengers did. Each team makes a line plot of each group's scores so they can compare the data from the two groups. Ultimately the question they address is, Which side won? The only rule is that they have to take into account the entire group's performance, not just the performance of the best player. When the teams report to the class, they will need to back up their decision with data.

Encourage students to look at the clusters of data, gaps in the data, and typical scores when making their comparisons.

### Reporting to the class: Who won?

Each team should report on its findings. As the teams present their results, it should become clear that different strategies for determining who won may lead to different winners. Expect students to give good reasons for choosing a particular strategy. They need to be clear about what winning means and to explain their thoughts about determining which group won.

If all the teams come up with decisions in their class's favor, ask:

If we were the challengers, is there any way we could argue that we won--and not this class? How would the data justify this different decision?

### Extension

Write a class letter to the challengers explaining who you think won and why. Include copies of the data. Ask them to examine the data and write back with their opinion about the winner.

Acknowledgements:

This activity is excerpted with slight modifications from Statistics: Middles, Means and In-Betweens, a unit of study for 5th and 6th grades from Used Numbers: Real Data in the Classroom. The Used Numbers curriculum units for grades K-6 were developed by TERC and published by Dale Seymour Publications. For more information contact Dale Seymour Publications, P. O. Box 10888, Palo Alto, CA, 94303, *800) 872-1100.

Author Info:

Dr. Susan N. Friel is Director of the Mathematics and Science Education Network, University of North Carolina, Chapel Hill. Dr. Jan Mokros is Co-director of the Mathematics Center at TERC. Dr. Susan Jo Russell is a principal author of the Used Numbers curriculum series and Co-director of the Mathematics Center at TERC.

Other Articles by Jan Mokros:

Bringing Math to Families

Children's Concepts of Average and Representativeness

Full Year Pilot Grades 3 and 4: Investigations in Number, Data, and Space

Where's the Math in Computer Games?

Other articles by Susan Jo Russell:

Changing the Elementary Mathematics Curriculum: Obsatcles and Challenges

Children's Concepts of Average and Representativeness

Developing Computational Fluency with Whole Numbers in the Elementary Grades