Activity 3: A Tool for Inquiry
Algebra with Manipulatives

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

how many squares?

The geoboard as a context for powerful ideas.

Lessons from Algebra: Themes, Tools, Concepts, by Anita Wah and Henri Picciotto.

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.

how long will it take?

30 minutes for additional work on the geoboard.

30 minutes for the reading.

Assignments:

Readings

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.

Hands-On
Activity 3
TO DO Checklist

__ Print this page!

Readings & Hands On
__ Complete the reading assignment.
__ Complete the hands on assignment.

Initial Thoughts
__ Post your initial comments on Activity 3 and read some of the module-based discussion in the Algebra Discussion area.

Local Study Group Meeting
__ Attend LSG meeting where colleagues share their new "tools" for inquiry.

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.

Reflection
There is no written assignment as you do the Hands On. However, while working with the materials think about the following:
  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?