What can I expect to get out of doing
this activity?

The geoboard as a context for powerful ideas.

In the previous activity, you laid a geometric foundation on the geoboard. While there was plenty of challenging work, the main focus was on access.

In this activity, you will build on that foundation and find that the geoboard microworld provides a context for a sophisticated discussion of important and difficult ideas in both geometry (the Pythagorean Theorem) and algebra (work with square roots, the distance formula).

You will also complete the reading you started last time on a tool-based pedagogy and an approach to algebra that is centered on thinking, rather than on the mechanical acquisition of skills.

30 minutes for additional work on the geoboard.

30 minutes for the reading.

A New Algebra: Tools, Themes, Concepts by Henri Picciotto and Anita Wah
(Journal of Mathematical Behavior, volume 12, #1, March 1993)

In the hard copy, read pp. 27-28, and part III, from the bottom of p. 37 to the end of the article. (The intervening text, pp. 29-37, is an optional extension.)

Optional: You may also want to take a look at a debate on geoboard use on Henri Picciotto's website. Here's the link.

Further Reflection
__ Visit the Algebra Discussion area again to read more of the dialogue and share your insights and questions.

Algebra: Themes, Tools, Concepts: pp. 238-239, #1, 3, 6, 9-11; pp. 331-332 #3-6. As you work through these problems, put yourself in the place of a student who does not know the Pythagorean Theorem or who knows it and does not think of using it. This will give you a better sense of how students typically do this.

If you prefer working on dot paper to using rubber bands on the geoboard, print out this page.

The main purpose of the activity is for students to develop their own understanding of square roots, distance in the Cartesian plane, and the Pythagorean Theorem.

1. What connections can you make between these lessons and other topics in mathematics? There are many possibilities.
2. These lessons are far from easy. For example, many teachers stumble on the very first problem. Is it appropriate to give students work that is so difficult that some teachers have trouble with it?