Abstract

This lesson allows students to examine tessellations and their geometric properties. This activity and discussions may be used to develop students' understanding of polygons and symmetry as well as their ability to analyze patterns and explore the role of mathematics in nature and our culture.

Objectives

Upon completion of this lesson, students will:

• have been introduced to tessellations
• have learned about polygons
• have identified types of symmetry in tessellations
• examined tessellations in the world around them

Standards

The activities and discussions in this lesson address the following NCTM standards:

Algebra

Understand patterns, relations, and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules

Apply transformations and use symmetry to analyze mathematical situations

• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations

• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

Links to other standards.

Student Prerequisites

• Geometric: Students must be able to:
  • Recognize regular polygons, such as triangles, rectangles and hexagons
  • Understand the difference between an edge and a corner
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• Access to a browser
• Pencil and Paper
• Regular Polygons Data Table

Key Terms

• polygon
• tessellation

Lesson Outline

1. Focus and Review

   Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

   • Has anyone ever heard of M. C. Escher? (Escher was a famous artist who enjoyed twisting perceptions of reality. He was responsible for works such as Reptiles, Horseman and many more that incorporated the use ofd tessellations.)

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

   • Today, class, we are going to learn about tellellations.
   • We are going to use the computers to learn about tessellations, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

3. Teacher Input

   • Introduce the tessellation applet in order to familiarize students to the idea of tessellations and how they developed.
4. Guided Practice

   • Have the students explore which regular polygons tessellate and why. Start them by examining tessellations of regular polygons including number of sides and interior angle measurements by using a data table. Encourage students to determine a pattern among the polygons that they tessellate. Ask the students to predict which regular polygons will and will not tessellate and why. Follow-up by having the students write a concise definition for a regular polygon tessellation. Have them expand this definition to describe a tessellation made from non-regular polygons.
   • After the students have determined which regular polygons tessellate, discuss the types of symmetry present in tessellations.
   • Have the students build tessellations and identify the types of symmetry present. Give them a table to record the basic shape used to tile and the types of symmetry present in the basic unit and in the tessellated pattern.
   • Discuss how angle measure, area, and perimeter apply to tessellations.
   • Allow students time to practice their knowledge about tessellations. Have teams of students work together. Instruct one student on the team to create a tessellation. Have that student describe the tessellation to other students and see if the other students can recreate the tessellation without looking. The students should formalize their terminology and describe the tessellation in terms of angle measure, polygon shape, symmetry, area and perimeter.
   • Lead a discussion about tessellations in the world. Ask students to identify tessellations that they see in their daily lives and in nature.
   • Discuss the ways that we perceive patterns. Lead a discussion about optical illusions to demonstrate how we perceive patterns. Also discuss the use of color in tessellations. Suggest that the students change the colors in their tessellations to see what effect that has on how they perceive the pattern. They may want to record their observations in a journal.

5. Independent Practice
   • Ask the students to use the tessellation activity to build tessellations of patterns they see in art and nature. You may also ask students to stretch the regular polygons into the letters of the alphabet or the letters of their name and tessellate the pattern. Have them record which polygon is best used to shape each letter. Also have them record what type of symmetries are present in each tessellation.
6. Closure

   • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outlines

This lesson can be rearranged in several ways. Here is an example of a shorter version:

1. The lesson can begin by introducing the tessellation applet to introduce students to the idea of tessellations and how they developed.

2. Discuss the types of symmetry present in tessellations.

3. Have the students build tessellations and identify the types of symmetry present. Give them a table to record the basic shape used to tile and the types of symmetry present in the basic unit and in the tessellated pattern.

Suggested Follow-Up

After these discussions and activity, the students should have practiced their ability to recognize symmetry in plane figures. Students can gain a deeper understanding of other pr inciples of geometry by exploring tessellations in the Geometry Lesson. The tessellation activity could also be used to explore spatial visualization and pattern recogni tion with the Visual Pattern Lesson.