Geometry in Tessellations

Abstract

This lesson allows students to examine tessellations and their geometric properties. The activity and discussion may be used to develop students' understanding of lines, planes, angles, and polygons.

Objectives

Upon completion of this lesson, students will:

• have been introduced to tessellations
• have learned about lines, planes, angles and polygons
• have experimented with the area and perimeter of polygons

Standards

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

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship

• 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

Student Prerequisites

• Arithmetic: Students must be able to:
  • identify polygons
  • measure angles
  • understand congruent figures
• 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
• Copies of supplemental materials for the activities:
  • Tessellations on Paper Worksheet
  • Regular Polygons Data Table

Key Terms

• tessellation
• polygon
• regular polygon

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:

   • Ask students if they remember what polygons are, and what makes regular polygons unique.

   • If needed, use the information in the discussion What are Tessellations?.

2. Objectives

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

   • Today, class, we will be talking more about the geometry involved in tessellations, and we will discover which types of polygons tessellate a plane.

3. Teacher Input

   Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

   • Open your browser to the Tessellate! page in order to demonstrate this activity to the students.

   • Show students how to select one of the regular polygon shapes and click the "tessellate" button to see it displayed.

   • Ask students to count the number of sides of the polygon.

   • Record the number of sides in the data table, and help students complete the rest of the information for the shape you have chosen.

4. Guided Practice

   Try another shape, letting the students take the lead in completing the data table for this new shape.

   • Encourage students to determine a pattern among the regular polygons that they work with. Ask the students to predict which regular polygons will and will not tessellate and why.

   • Select the third regular polygon, observe what it looks like in the Tessellate activity, and then complete the data table for this shape.
     • Ask students what the data table would look like for 5, 7 and 8-sided polygons.

     • Help students analyze the data and draw a conclusion about which shapes will tessellate the plane and why.

5. Independent Practice
   • Print out copies of the Tessellations on Paper Worksheet and distribute them to students.

   • Alternatively, have the students complete a similar exercise on the computer using the Tessellate activity, a screen capture utility and a drawing program.

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 if there is only one available computer:

• You can model all the parts of the lesson for the students, asking them to complete the data table with you as a class.

• Model the connect the dots activity to the whole class on the computer using the Tessellate activity, a screen capture utility and a drawing program, but then distribute the Tessellations on Paper Worksheet to each student to complete individually.

Suggested Follow-up

• Symmetry in Tessellations
• Visual Patterns in Tessellations