# Histogram Sorting Using Cooperative Learning

This material was originally developed through CAUSE

as part of its collaboration with the SERC Pedagogic Service.

#### Summary

Using cooperative learning methods, this activity provides students with 24 histograms representing distributions with differing shapes and characteristics. By sorting the histograms into piles that seem to go together, and by describing those piles, students develop awareness of the different versions of particular shapes (e.g., different types of skewed distributions, or different types of normal distributions), and that not all histograms are easy to classify. Students also learn that there is a difference between models (normal, uniform) and characteristics (skewness, symmetry, etc.).

## Learning Goals

This activity has two major goals: (1) to give students experience with a variety of histograms of data, and (2) to help them better recognize different shapes and characteristics. Too often students only see one or two perfect examples (e.g., normal, right skewed) and have a difficult time describing and classifying histograms of real data. This activity also helps students determine which characteristics can appear together (e.g., skewed and bimodal) and which cannot be used together to describe a distribution (e.g., skewed and symmetric). This activity may be used to help students better understand the relationship between descriptions of data sets and the graphs that could be created from these data sets.

## Context for Use

This activity is usually done at the beginning of a unit on distribution, takes about 30 minutes, and may be used with any size classroom. The activity may be easily adapted for junior high, high school, and college-level instruction.

## Description and Teaching Materials

This activity uses sets of 24 histograms, available on the STAR library on CAUSEWeb. The histograms are located under the title "Student's version" (upper lefthand corner) in either HTML or Word format. One set of graphs is needed for each group of students doing the activity. The pages need to be cut so that only one graph is on a piece of paper. These graphs can then be placed in an envelope or clipped together.

The following website can be used for a follow-up debriefing activity: http://web.archive.org/web/20170328052252/http://www.tc.umn.edu/~delma001/stat_tools/. Detailed lesson plan of how to use cooperative learning for this activity. (Acrobat (PDF) 32kB Jun30 06)

The following website can be used for a follow-up debriefing activity: http://web.archive.org/web/20170328052252/http://www.tc.umn.edu/~delma001/stat_tools/. Detailed lesson plan of how to use cooperative learning for this activity. (Acrobat (PDF) 32kB Jun30 06)

## Teaching Notes and Tips

Cooperative Learning:

See "Histogram Sorting Using CL Lesson" file for specific directions about teaching this lesson using Cooperative Learning strategies.

Time Involved:

- 5 minutes to introduce the activity

- 10-15 minutes for students to work in groups, sorting graphs

- 10 minutes for instructor-led discussion of graphs

- 5 minutes for follow up questions

Practical Tips:

- Each group of students will need one set of histograms. Make sure

you have enough sets of histograms for the number of groups in

your classroom.

- The activity works best BEFORE students have formally studied different shapes of graphs.

- Typically, students will sort their graphs into the following categories: uniform, normal, skewed, and bimodal. There may also be some smaller groupings such as right skewed and left skewed.

- Students often find the uniform and normal graphs easiest to sort. The also find unimodal graphs easier to classify than bimodal graphs. Students have more difficulty with the graphs that are skewed and bimodal.

- Throughout the lesson, instructors should encourage students to consider why it is important to describe a distribution.

- After students complete their sorting, the instructor can lead a class discussion using questions such as:

o What was the easiest group to sort? Which graphs are in that group?

o How many different groups did you find? Which graphs are in each? What did you call them? What features did they have in common? Etc.

o Which graphs were hardest to sort or classify? Why?

- As students suggest their categories, the instructor may also want to refer to graphs on the following website:

http://web.archive.org/web/20170328052252/http://www.tc.umn.edu/~delma001/stat_tools/.

This applet includes most of the graphs in the activity. There are buttons along the bottom that represent five different categories of distributions. When you click one, it brings up the set of graphs with that type of characteristic. You use the PREV and NEXT buttons on the right to view the graphs in each set.

- The correct statistical terms for the graphs (uniform, normal, right and left skewed, bimodal) can be introduced if students have not yet learned these terms. Models (uniform, normal) can be described in terms of symmetry and shape (bell shape or rectangular). Other distributions that don't fit these models can be described in terms of their characteristics (skewness, bimodality or unimodality, etc). A discussion of which descriptors can and cannot go together may follow.

- These points may be included in the discussion of graphs following the activity:

o Ideal shapes: density curves vs. histograms

o Different versions of ideal shapes

o Idea of models, characteristics of distributions

o Statistical words vs. descriptors

o Normal, skewed, uniform, bimodal, symmetric: which can be used together?

See "Histogram Sorting Using CL Lesson" file for specific directions about teaching this lesson using Cooperative Learning strategies.

Time Involved:

- 5 minutes to introduce the activity

- 10-15 minutes for students to work in groups, sorting graphs

- 10 minutes for instructor-led discussion of graphs

- 5 minutes for follow up questions

Practical Tips:

- Each group of students will need one set of histograms. Make sure

you have enough sets of histograms for the number of groups in

your classroom.

- The activity works best BEFORE students have formally studied different shapes of graphs.

- Typically, students will sort their graphs into the following categories: uniform, normal, skewed, and bimodal. There may also be some smaller groupings such as right skewed and left skewed.

- Students often find the uniform and normal graphs easiest to sort. The also find unimodal graphs easier to classify than bimodal graphs. Students have more difficulty with the graphs that are skewed and bimodal.

- Throughout the lesson, instructors should encourage students to consider why it is important to describe a distribution.

- After students complete their sorting, the instructor can lead a class discussion using questions such as:

o What was the easiest group to sort? Which graphs are in that group?

o How many different groups did you find? Which graphs are in each? What did you call them? What features did they have in common? Etc.

o Which graphs were hardest to sort or classify? Why?

- As students suggest their categories, the instructor may also want to refer to graphs on the following website:

http://web.archive.org/web/20170328052252/http://www.tc.umn.edu/~delma001/stat_tools/.

This applet includes most of the graphs in the activity. There are buttons along the bottom that represent five different categories of distributions. When you click one, it brings up the set of graphs with that type of characteristic. You use the PREV and NEXT buttons on the right to view the graphs in each set.

- The correct statistical terms for the graphs (uniform, normal, right and left skewed, bimodal) can be introduced if students have not yet learned these terms. Models (uniform, normal) can be described in terms of symmetry and shape (bell shape or rectangular). Other distributions that don't fit these models can be described in terms of their characteristics (skewness, bimodality or unimodality, etc). A discussion of which descriptors can and cannot go together may follow.

- These points may be included in the discussion of graphs following the activity:

o Ideal shapes: density curves vs. histograms

o Different versions of ideal shapes

o Idea of models, characteristics of distributions

o Statistical words vs. descriptors

o Normal, skewed, uniform, bimodal, symmetric: which can be used together?

## Assessment

The Cooperative Learning lesson plan emphasizes several ways that instructors may incorporate positive interdependence (i.e., collective assessment) and individual accountability. See the Cooperative Learning module.

A sample assessment is included with the original activity. This assessment can be used to assess students' ability to correctly describe graphs and understand the difference between graphs.

A sample assessment is included with the original activity. This assessment can be used to assess students' ability to correctly describe graphs and understand the difference between graphs.

## References and Resources

The following website provides graphs and categories that students might use to create their categories for sorting histograms in part one: http://www.tc.umn.edu/~delma001/stat_tools/

CAUSEweb includes the original version of of this activity