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The participants' "experiences as learners" on Tessellation and/or Area are intended to contribute to developing a first "image" of the nature and contents of our illustrative units. However, we believe that this by itself is not sufficient to translate these experiences into practice in a middle school mathematics classroom. Teachers may still question how middle school students (rather than adult learners) would actually respond to these unusual learning experiences, how activities could be scheduled within the constraints of their own mathematics courses and how much time they may take, or even how to word certain tasks. In addition, we also thought it would be important for the participants to clearly identify the instructional goals informing these units (since several of them are quite different from what they may have been used to) and to think about important "mathematical ideas" embedded in the units.

It is important to note that most published "instructional materials" do not provide teachers with this kind of information. At the time we attempted to create such supporting materials for our two illustrative units, we did not have many models we could refer to and had instead to try to develop some new formats. In the creation of these materials, we devoted considerable attention not only to what kind of information the participants needed to have about our illustrative units, but also how such information could be "packaged" so that the participants could be truly supported in their planning of such novel instructional experiences, and yet be encouraged to make important instructional decisions so as to make the unit "their own". This presented a very challenging task and resulted in the creation of an unusual set of supporting materials which has now been collected in the booklets entitled "Investigating tessellations to learn geometry: Supporting materials for teachers" and "Understanding area through investigations: Supporting materials for teachers" (enclosed in this package).

Each of these booklet contains the following sections:

• A "message to the reader." With the goal of orienting teachers to the information provided by these unusual supporting materials, in this first section we briefly describe the contents and organization of the booklet, and provide suggestions about how to read/use specific sections.
• A brief description of the scope and instructional goals of the unit. We think it is important to articulate upfront the goals assumed for our units, especially since they differ considerably (in content as well as priorities) from those implicitly assumed in most traditional mathematics instruction. The teacher's clear understanding of these goals and of their major instructional implications is indeed necessary in order to implement the units in the spirit they were intended, develop appropriate expectations in terms of classroom dynamics and students' responses, and be able to reasonably evaluate the effects of the unit on students' learning.
• An overview of the major components of the unit and their rationale. This overview is intended to provide teachers with an understanding of the structure of the unit and its main components so that they can gain a "big picture" of the proposed unit, and use that to guide their reading of the remaining parts of the booklet.
• A discussion of key mathematical ideas addressed in the unit. Far from providing a typical "math textbook" treatment of the topic covered in the unit, this section is intended to be raising questions and issues that should invite teachers to rethink the mathematical content that could be covered within the unit and what students could learn from it. As such, reading this section is likely to provide any mathematics teacher (regardless of their math background) with valuable new perspectives and/or information.
• A "detailed flexible plan." Although this section is intended to directly support the planning of the unit, it does not provide teachers with "lesson plans." Rather, the goal of this section is to both articulates in more detail the characteristic features of each component of the unit, and suggest a variety of possible activities that teachers could include in their plan (presented in the form of "options" in terms of both content and pedagogy). In other words, this could be thought of as an organized compendium of various successful implementations of the unit by different teachers in different instructional settings. Unfortunately, the attempt to incorporate a rich collection of alternatives within a unique plan has the drawback of making this format somewhat cumbersome; thus, participants may need some explicit suggestions about how to "read" this document effectively.
• A few brief accounts of classroom implementations of the unit accompanied by selected artifacts. First of all, these reports are intended to provide images of the diverse "real-life" units that resulted from implementing our illustrative units in a variety of instructional contexts. These accounts also provide examples of specific assignments and/or hand-outs, as well as information about the different choices by specific teachers in terms of sequence and timing of various activities. It is important to note that these experiences as not offered as a model to be "copied", but rather as an inspiration and a source of ideas that could be further modified and/or adapted for each teacher's own classroom situation.

The previous documents are also complemented by a more "in-depth look" at a classroom implementation of the Tessellation and Area units -- provided in the form of the video "Exploring tessellations to learn geometry: A classroom experience" and the detailed "story" included in the booklet "Understanding area through investigations: Supporting materials for teachers". These two formats combined are intended to provide the participants with a rich image of the classroom dynamics, teacher's choices and students' responses that could occur in a mathematics classroom informed by an inquiry approach and including a diverse student population.

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