### Purpose

To look at the questions that can be posed and answered by examining data distribution. To look for circumstances that might bias the results of a study.

### Context

In this investigation, students will analyze data collected by themselves and other students.

Students will create data distributions in order to determine the highest, lowest, and middle values, as well as pile ups and gaps, for a particular set of data. Students also will be introduced to the notion of sample size and how it can affect results. Finally, students will begin to explore the questions that can be answered based on a set of data, and the additional questions it might prompt one to ask.

Research suggests that it is premature to introduce the terms mean, median and mode at this grade level. (*Benchmarks for Science Literacy*, p.354.)

### Motivation

Begin this lesson by conducting a class survey to determine how many family members each student has. Students can record the number of family members on a Post-It® note, then line up the Post-Its® from smallest to largest value along a number line on a chalkboard or wall. Have students examine the data collection.

Ask students:

- Are there places where the data seem to pile up?
- Are there places where there is little or no information?
- What number is in the middle?
- What are the highest and lowest numbers?
- What question(s) can we answer by looking at this?
- What other questions might the data lead us to ask?
- Can we answer these questions based on the data that we have? Why or why not?

### Development

Have students use their What Can Data Tell Us? student esheet to go to and play Tower of Hanoi from the Math is Fun. In this game, students must transfer a stack of disks between three poles, moving only one disk at a time. Students must reassemble the entire stack of disks on a new pole, without allowing a larger disk to rest on top of a smaller one at any time.

** Note:** You may wish to give students hints or strategies, or play the game once as a class, in order to familiarize students with the game. Students have the option of choosing using from three to six discs.

Give students ample time to solve the puzzle, working alone or in pairs. Once students have successfully completed the task with three discs, have them record the number of moves it took to solve the puzzle on the What Can Data Tell Us? student sheet. Then have them play the game again using four, again recording the number of moves that it took to solve the puzzle.

Have students submit their results on Post-it® notes. Tell students that they will analyze the class data. Have three volunteers arrange their Post-its® from smallest to largest value across a blackboard or wall.

Ask students to look carefully at the data distribution and record student responses to these questions:

- Are there places where the data seem to pile up?
- Are there places where there is little or no information?
- What number is in the middle?
- What are the highest and lowest numbers?
- How many students' results are included in our data distribution? How do their results compare to yours?

Allow the rest of the class to add their Post-its® to the data distribution. Have students answer the same questions using the larger sample:

- Are there places where the data seem to pile up? Has this changed?
- Are there places where there is little or no information? Has this changed?
- Where is the middle? Has the middle changed?
- What are the largest and smallest values? Have these values changed?
- How much is the data spread on both sides of the middle? Has this changed?

Introduce the concepts of sample and bias by asking these questions:

- How many students' results did we look at previously? How many are we using now? How might the size of a sample affect our results?
- Is there anything else that might influence our results?

For an online tutorial that reinforces the concepts of sample size and data analysis, have students use their student esheet to go to PlaneMath's Experimental Department. Here, students learn how to analyze a collection of data. While the entire tutorial is a bit unwieldy for young students, the sections pertaining to Sample Size and Median would be useful for reinforcing concepts presented earlier in the lesson. One or more of the quizzes also might be suitable for a review or homework assignment.

Using their student esheet, students can visit the Cereal Box Problem Simulation from the Office of Mathematics, Science, and Technology Education at the University of Illinois at Urbana-Champaign. Although the project is designed for older students, the simulation can be used to provide students with data that can be graphed and/or displayed as a data distribution.

To begin, present students with the following problem: There are three different prizes that come in a cereal box. How many boxes of cereal would we need to buy in order to get all three prizes?

Tell students that their job will be to collect data and create a data distribution for the Cereal Box Simulation.

Have students record their answers to these questions:

- What will your sample size be? Why?
- How will you record your results?
- How will you analyze your results?
- What questions will you be able to answer by looking at this data?

Have students run the online simulation and record results. Have them work independently to display their results along a number line and answer these questions:

- Are there places where the data seems to pile up?
- Are there places where there is little or no data?
- What number is in the middle?
- What are the highest and lowest numbers?
- What is your sample size?
- What question(s) can be answered by looking at this data collection?

### Assessment

Allow students time to discuss and compare their results. Collect and assess students' responses to the discussion questions in this lesson.

### Extensions

Students can build and test paper airplanes or gliders according to the methods outlined in PlaneMath's Experimental Department. Instructions for building a Straw Glider also can be found on the PlaneMath site. Students can measure each flight, then organize and analyze the data on a number line.

Students can work in pairs or small groups to conduct a class survey related to student height, number of teeth lost, number of pets, number of siblings, monthly attendance, etc. They can present the results of the survey to the class, using either a graph or number line.