# Reasoning About Center and Spread: How do Students Spend Their Time?

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### Summary

This activity is designed to develop student reasoning about variability in data sets by having them make predictions about the average number of minutes (in a 24 hour period) they spend traveling to and from school, exercising, eating, studying, and talking on the cell phone.

The students predict which of these daily activities will have a lot and which will have a little variation. Then the students examine their choices through the use of computer software. The final task in the activity has the students use reasoning about distributions to examine graphs and summary statistics from real world data to choose variables that have a lot and have a little variability.

## Learning Goals

To promote student reasoning about center and spread through the use of conjectures, computers and real world data.

## Context for Use

This activity is usually done at the end of a unit on descriptive statistics in an entry level statistics course. The lesson takes approximately 75 minutes and is well suited to any size classroom. The activity may easily be adapted for junior high, high school, and college-level instruction.

## Description and Teaching Materials

The students will use graphs and statistics to reason about center and spread. Further, they will test their conjectures about the amount of time students spend traveling to school or exercising in a week by first making predictions and then testing these predictions against real world data.

This activity asks questions to promote student reasoning about center and spread and has the students use graphs and statistics to analyze their student daily time data. The worksheet for reasoning about center and spread consists of having the students make predictions about the amount of the time the students in their class spend: traveling to school, exercising in a week, communicating with their parents, eating, on the internet, studying, and on their cell phones.

- A copy of the lesson plan as a word document can be accessed by clicking the link below.

Lesson Plan (Microsoft Word 30kB Mar1 07)

**Instructor Lesson Plan**

This lesson builds ideas of variability in distributions, tying together the concepts of shape, center and spread.

**Goals for the Lesson:**

- Review the concepts of distribution, center, and spread.
- Understand how the concepts of distribution, center and spread are related.
- Know when to use each type of measure of center and variation.
- Be able to reason about amounts of variability in data sets, particularly when comparing different variables measured on the same scale.

**Materials Needed:**

- Student Worksheet: For a copy of the student worksheet (word document) click the link text below.

Student Worksheet (Microsoft Word 48kB Oct20 06)

- Data File of Time data: Have the students either collect data in class or have previously collected data available as part of a student survey.

**How to Guide Students to Make and Conjectures**

- Have students predict which daily activities (as outlined in the worksheet) will have a lot and which activities will have a little variation.
- Have students explain their predictions and reasons, as well as graphs of their predicted distributions.
- Have the students examine use computer software to analyze class data for these variables.
- Have students, working in small groups, examine graphs and summary statistics as they reason about the shape, center and spread of the entire set of variables and choose which variables have a lot and which variables have a little variability.
- Have students explain and justify to the entire class their conclusions.

**The Process of Having Students Make and Test Conjectures**

- Begin with a discussion about why variability is such an important concept in statistics and ask the students what variability means to them.
- Divide the students into groups of 3 or 4 and have them complete the first, second and third task on Reasoning About Center and Spread Worksheet.
- Compare and discuss what students predicted and what they found out. Ask them why they chose the variables they did and what criteria they were using to make their choices.
- As a class, collect data from students in this class about the typical number of minutes per day for each activity listed at the beginning of their worksheet. This may be done prior to class, as part of a first day class survey. Enter the class data or access a file with these data if previously entered.
- Have the students compare their predictions to the actual distributions by analyzing class data.
- Ask students how they did and if their predictions were correct. For example, which predictions they did better with (a lot or a little variation)? Their predictions of center, what their shapes of the distributions looked like, and did the shape play a part in how well they did on their predictions?
**For wrap up:**Have a whole class discussion about which variables the students believe have the most and which variables have the least amount of variation, why they chose those variables, which measure of variability they looked at to make their selection, which variable has the largest or smallest range, IQR, standard deviation, what each measure of variation tells us, how do they relate to measures of center and shape of the distribution, and how can we best determine a lot or a little variability?

## Teaching Notes and Tips

Time Involved:

- 5 minutes to introduce the activity
- 20-25 minutes for students to work in groups
- 10 minutes for instructor-led discussion throughout the activity
- 5 minutes for follow-up questions

Helpful Hints:

Be sure to ask students to share their predictions and reasoning first in small groups, then with the entire class. Ask probing questions such as, why did you think that. Ask representatives from different groups to report on their results.

For more tips on using cooperative groups, see the Cooperative Learning Module.

## Assessment

- Ask students to conjecture about other variables, measured on the same scale, that have a lot or a little variability.
- Have students think about test scores for different sections of a calculus class, and what a lot or a little variability would mean in terms of student performance on these tests.
- Why might one class have scores with a lot of variability?
- Why might another class have scores with a little variability?

## References and Resources

**Garfield, J., delMas, R., and Chance, B. (in press)**. Using student informal notions of variability to develop an understanding of formal measures of variability. In *Thinking about Data*, edited by P. Shah and M. Lovett.