INTRODUCTION AND LESSON
ONE
Introduction: Why is this lesson constructivist?
A constructivist lesson is rooted in an authentic context that draws upon the students' realworld experience. Such a lesson is especially beneficial for teaching abstract mathematical concepts, as illustrated in the lessons below, which are designed to teach the geometric concept of tessellations. Rather than presenting the definition of tessellations directly to the class, these lessons are structured to facilitate the students' discovery of the concept through a series of teacherguided constructivist activities. Students construct their understanding of tessellations by using Webbased resources, participating in multimedia activities, and applying their knowledge to a realworld problem.
Lesson One encourages students to formulate a definition of tessellations by comparing and contrasting different patterns and shapes of real objects provided by the teacher  that is, crackers. The discussion prompts the students to probe the various aspects of the relationship between shapes and patterns as well as their function in the real world.
The constructivist classroom also seeks to examine a problem or phenomenon through multiple perspectives. Lesson Two integrates Webbased resources that introduce students the world of art and culture. The works of M.C. Escher, for example, provide a visual representation of tessellation through another perspective, the medium of art. This encourages students to draw on their previous insights about crackers to identify more complex patterns of tessellation in visual representations. Interactive multimedia programs provide "handson" experience as students construct their own tessellations. This transforms an abstract concept into a concrete process in which students engage directly, thereby enhancing their understanding and knowledge.
Finally, the constructivist classroom transfers academic learning to the real world. Lesson Three presents a problem that the class must solve by applying their knowledge and understanding of tessellations. The students' task is to design a suitable package for a cracker manufacturer, taking into account certain variables such as the shape of the crackers, cost, efficiency, limited budget, and ecological considerations.
All of these activities take place through a collaborative effort, as students work in groups. This allows students to learn from each other through the exchange of ideas and knowledge. It also enhances and promotes communication skills, since students learn to convey their ideas clearly and effectively.
Shape Shifters
Overall Unit Question: What are tessellations?
Age Range: Middle school
Standards Addressed:
 Students explore the relationships among geometric transformations (translations, reflections, rotations, and dilations), tessellations (tilings), and congruence and similarity.
 Students investigate and describe geometry in nature and realworld applications.
 Students design a tessellation.
Computer Resources
You will need at least one computer with Internet access to complete
this lesson. While many configurations will work, we recommend:
Modem: 56 Kbps or faster.
Browser: Netscape Navigator 3.0 or above or Internet Explorer 3.0 or above.
Macintosh computer: System 7.0 or above and at least 32 MB of RAM.
PC: Pentium processor with at least 32 MB of RAM, running Windows 95 or higher.
Plugins: The animation sites require a Java capable browser, available with Netscape 3.0 and higher and Explorer 3.0, or Shockwave, available for free download at http://www.macromedia.com. Java applets may take some time to load. Remind your students to be patient.
Lesson One: Discovery
How can tessellating shapes help a cracker manufacturer make a profit?
1. Bring in various boxes of crackers. Set the classroom up like a supermarket. Place the products on desks like aisles so that the students can browse through. Ask the students to analyze the shapes of the crackers and take notes about what they find. The class might start identifying similarities among the shapes of the crackers. After the students have had a chance to generate some patterns, you might categorize the crackers into "tesselating" and "nontesselating" and see if the students can generate possible definitions by analyzing the crackers in the two categories. Ask students to determine a definition of tessellation by looking at different patterns of crackers. Why are tesselating patterns so common in these food products? Can you think of a way this kind of pattern benefits the cracker manufacturer?
