MERLOT Mathematics Portal Visit Main MERLOT

# Pythagorean Theorem Investigations

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

#### Summary

Secondary Math Ed majors or Geometry students are directed to a Web site that contains a rich and extensive collection of proofs of the Pythagorean Theorem, some accompanied by Java applets. Each student then selects two proofs to prepare to present to the class (a geometric proof and a visual proof) and writes a detailed outline of each proof. The student also indicates one additional proof that was of particular interest and the reason(s) for its selection.

In a subsequent class, students are selected to make their presentations, either individually or in teams; the number of presentations may be limited due to time constraints and can be selected by lottery or a similar system.

## Learning Goals

To enable students to:
• gain a deeper understanding of the Pythagorean theorem and its rich history
• recognize the variety of ways in which theorems can be proved algebraically, geometrically and visually
• develop their understanding of the nature of mathematical proof
• exercise their classroom presentation skills

## Context for Use

This activity can be used in a Geometry class as an introduction to the nature of geometric proof or in conjunction with the study of the Pythagorean Theorem or in a Secondary Math Ed class to develop understanding of proof and to exercise classroom teaching skills.

The activity will require approximately 60 minutes outside of class depending on how long students spend on the interactive Java applets and 45-60 minutes or more in class, depending on how many presentations are given. Each student indicates his or her additional recommendation for an interesting proof. The instructor can compile this list of recommendations and share the results with the class.

## Description and Teaching Materials

Activity description
• By means of an assignment sheet (Rich Text File 15kB Jul9 07), students are directed to read through a Web-based collection of Pythagorean Theorem proofs outside of class and select two from the first ten to prepare to present to the class. In addition, each student is directed to select an additional proof that was of particular interest to them.
• During a subsequent class, students are selected to make their presentations, either individually or in teams. Depending on the nature of the class, the instructor may lead a discussion of the presentation techniques (Secondary Math Ed students) or of the nature of the various proofs presented (Geometry students). The instructor may also compile a list of the additional recommended proofs to share with the class.

## Teaching Notes and Tips

The collection of proofs at Cut-the-Knot is both rich and extensive. Students will have to read thoroughly and at times will have to carefully label the diagrams and verify the statements in the proof to their own satisfaction in order to clearly understand the proofs. The visual proofs lend themselves to the use of colorful manipulatives.

As part of the post-presentation discussion, the three short video segments (approximately 20 seconds each in length) indicated in the References and Resources below can be shown to the class; the narrated animations are particularly effective in illustrating the proofs.

## Assessment

Presentation assessment will depend on the nature of the class and on the extent of the presentations.

## References and Resources

Project Mathematics–Pythagorean Theorem
Several narrated and animated proofs of the Pythagorean theorem are presented in short video segments. In particular, segments #9, #10 and #11 provide insightful animations that relate to some of the first ten proofs shown at Cut-the-Knot-Pythagorean Theorem listed above.
MERLOT description of the Project Mathematics–Pythagorean Theorem site
Direct link to the Project Mathematics–Pythagorean Theorem site