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 Addressed

Grade 3

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 4

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 5

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

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.

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.

Third Grade

  • Geometry

    • Reason with shapes and their attributes.

Geometry

  • Congruence

    • Experiment with transformations in the plane
    • Understand congruence in terms of rigid motions

Grades 6-8

  • Geometry

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

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

  • Geometry and Measurement

    • COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts 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.

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.

Integrated Mathematics III

  • Geometry and Measurement

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

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
    • 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.

Geometry

  • Geometry

    • Standard G-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

Textbooks Aligned

Grade Six

  • Shapes and Designs

    • Investigation One: Bees and Polygon

7th

  • Module 8 - Heart of the City

    • Section 4: Quadrilaterals and Tessellations
      • Reason for Alignment: The Geometry in Tessellations lesson is another look at the properties of tessellating polygons. This one is fun for the students as well as a quick way of investigating these figures.

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 for experimenting with the activities.

Teacher Preparation

Key Terms

polygon

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

regular polygon

A polygon whose side lengths are all the same and whose interior angle measures are all the same

tessellation

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

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.

  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 Outline

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.