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

have examined tessellations in the world around them

Standards Addressed:

Grade 10

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 3

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 4

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 5

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 6

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 9

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Geometry

Congruence

Experiment with transformations in the plane

Understand congruence in terms of rigid motions

Third Grade

Geometry

Reason with shapes and their attributes.

Grades 6-8

Geometry

Apply transformations and use symmetry to analyze mathematical situations

Use visualization, spatial reasoning, and geometric modeling to solve problems

Grades 9-12

Geometry

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

Apply transformations and use symmetry to analyze mathematical situations

Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

Data Analysis and Probability

Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

Geometry and Measurement

Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Integrated Mathematics III

Geometry and Measurement

Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Technical Mathematics I

Geometry and Measurement

Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

Geometry and Measurement

Competency Goal 1: The learner will use properties of geometric figures to solve problems.

3rd Grade

Geometry

The student will demonstrate through the mathematical processes an understanding of the connection between the identification of basic attributes and the classification of two-dimensional shapes.

5th grade

Geometry

The student will demonstrate through the mathematical processes an understanding of congruency, spatial relationships, and relationships among the properties of quadrilaterals.

6th Grade

Geometry

The student will demonstrate through the mathematical processes an understanding of shape, location, and movement within a coordinate system; similarity, complementary, and supplementary angles; and the relationship between line and rotational symmetry.

7th Grade

Geometry

The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the

4th Grade

Geometry

4.17.c The student will investigate congruence of plane figures after geometric transformations such as
reflection (flip), translation (slide) and rotation (turn), using mirrors, paper
folding, and tracing.

Geomety

4.17.c

5th Grade

Geometry

5.15a The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize, identify, describe, and analyze their properties in order to develop definitions of these figures

6th Grade

Geometry

6.14 The student will identify, classify, and describe the characteristics of plane figures, describing their similarities, differences, and defining properties.

6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of noncongruent and congruent figures will be included.

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 for experimenting with the activities

A closed plane figure formed by three or more line segments that do not cross over each other

tessellation

A tessellation is a repeated geometric design that covers a plane without gaps or overlaps

Lesson Outline

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 of tessellations.)

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.

Teacher Input

Introduce the
t=Tessellation Applet in order to familiarize students to the idea of tessellations and how they developed.

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.

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.

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 Outline

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

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

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.

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.