What is the reason for this reflection?

In the hands-on activity, you have explored the area of squares on the geoboard. Please comment on your new experiences with this tool.

In the reading, you have seen how the authors of this activity place the geoboard in the broader context of the teaching of important concepts in algebra and geometry and you have been exposed to their vision of a new algebra.

Share your insights and your problems with the members of your virtual group and the Field Expert. (One Field Expert will be reading the posts and representing this curriculum.)

These questions focus on problem #1, p. 238.

a. This initial problem is surprisingly difficult for many students. At first, they may believe that there are only ten possibilities, because they are only considering answers with horizontal and vertical sides. Later, they may think that the only additional squares are the ones that make a 45° angle with the horizontal. This is very much an activity where group effort is required. Even when they work in groups, students may need hints. What are the causes of this visual block? How can one help students get around it?

b. What was your strategy for finding more than fifteen solutions? (At least two major approaches tend to surface in the classroom, one based on the slope of the sides, the other on connecting well-chosen points on the sides of horizontal-vertical squares.)

c. Did you find 35 squares? If so, two of them are repeats of others. How would you find out which ones? Comment.

d. Is it important for students to find all 33 solutions? How much time should be allotted to this initial exploration?

e. What additional questions arise in the course of working on this problem?

Read the comments of others and make two of your own in the Algebra Discussion Area. Look for the Tools: Algebra Activity 3 thread.

What is the reason for this second reflection?

In the previous reflection, you were encouraged to give a lot of thought to the problem of finding the geoboard squares. In this round, you will:

one hour to post your ideas.

one hour to comment on postings and other reflections.

Please make TWO comments on any TWO threads below: CONNECTIONS, PEDAGOGY, or VISION

1. Mathematical connections: this activity is rich with potential connections -- some of them are outlined here, and more can be found in the extensions below. Comment one ONE entry that interests you.
   • The concept of slope is helpful in creating squares.
   • If students can find squares and their areas, they can calculate the distance between any two geoboard points. This is an alternative method to the distance formula, which can be applied without knowing it, and even without knowing the Pythagorean Theorem.
   • How does the introduction to the Pythagorean Theorem on page 331 compare with or connect to other approaches? Does it promote better understanding of the theorem?
2. Pedagogy: React to ONE of the following thoughts from the author of the geoboard lessons:
   • Student-devised strategies to find geoboard areas can give students access to powerful and profound ideas.
     The geoboard, much more than chalk and worksheets, provides the necessary scaffolding to support the students' own thinking at every step of the way.
   • The teacher role is more important than in a traditional instructional situation. At the limit, the teacher in the traditional format can be replaced with a book or a video, but here the teacher's role as coach to group discovery and discussion is essential.
   • Selection of appropriate problems and activities is essential also. They need to fall within the students' area of competence, but also to stretch them to take the next step, but not so difficult as to freeze students out.
3. Vision: Comment on the vision of a new algebra course that was expressed in the reading. Comment one ONE entry that interests you.
   • Is it desirable? Is it realistic? Could such an approach be implemented in your school district?
   • What would be the main obstacles?
   • Many new "reform minded" textbooks include real world applications ("themes" in the language of the reading). They also feature increased use of technology, especially the graphing calculator. Yet the tool-based approach as outlined in the reading is not particularly widespread. For example, few high school teachers use manipulatives of any type. Why is that?

Read the comments of others and make two of your own in the Algebra Discussion Area. Look for the Tools: Algebra Activity 3 thread.

For geoboard lessons on square roots, see pages 239 (problems #12-15) and 279 of the handout.

If you want to explore the application of geoboard squares to the concept of distance, see pp. 279-280, 328-329, and 332.

For a geoboard representation of the equivalence 2xSQRT(5) = SQRT(20), and related work on radicals, see pp. 334.

Finally, for a review of all these topics, plus connections between slope, the Pythagorean Theorem, and simple radical form, see the rich activity on page 339.

If you pursue any of these extensions, consider sharing what you find out with others in the discussion area.