Name: Lynda Kozicz Leahey
School District: Absegami High School Absecon, New Jersey
Title of Lesson: Transformations and Tessellations
Subject: Geometry
Category: 9 to 12
Additional Materials


Goal and Objectives of Lesson

I. The goal of this unit is to introduce students to the concepts of transformations and tessellations. They will explore the attributes of transformations: reflections, translations, rotations and dialations by using pattern blocks, algebraic notation and computer design. They will utilize these concepts to construct a tessellation.

New Jersey Core Curriculum Content Standards
4.2 All students will communicate mathematically through written oral, symbolic, and visual forms of expression.
4.3 All students will connect mathematics to other learning by understanding the interrelationships of mathematical ideas and the roles that mathematics and mathematical modeling play in other disciplines and in life.
4.5 All students will regularly and routinely use calculators, computers, manipulatives and other mathematical tools to enhance mathematical thinking, understanding, and power.
4.7 All students will develop spatial sense and an ability to use geometric properties and relationships to solve problems in mathematics and in everyday life.
4.11 All students will develop an understanding of patterns, relationships, and functions and will use them to represent and explain real-world phenomena.
4.12 All students will develop an understanding of algebraic concepts and will use them to represent and analyze relationships among variable quantities and to solve problems.

Summary of the Lesson and Detail of Activities

To find the image of a figure after a reflection, translation, rotation and dialation.

Method
Students will reflect a design of pattern blocks over a line of reflection. When doing so, they will observe the symmetry of the reflection. In addition, students will reflect a design of pattern blocks over a set of intersecting lines and observe that this series of reflections results in a rotation.

Students will graph reflections and translations on the coordinate plane. They will develop an algebraic expression to determine the transformation. (x,y) à (-x, y) is a reflection over the y axis, and (x,y) à (x, -y) is a reflection over the x axis.
By adding or subtracting a constant from either coordinate, students will observe that this creates a translation. Thus the rule, (x,y) à (x+2, y-3), would yield a translation two units to the right and three units down.
Students will enlarge and reduce a figure by using ratio, proportion and one point perspective drawings.

II. To construct a tessellation using translations.
Students will construct a tessellation using computer software, Paint on Windows 95. By applying cutting and pasting techniques, students will create an Escher Art type design of translations that will completely fill the plane. They can embellish their design by using the paintbrush tools and their own creativity.
Students will work in pairs or alone depending on the number of computers available. The directions for this section are given on a handout.

III. To construct a regular tessellation using translations and rotations. Students will construct a regular tessellation using the Draw toolbar on the software Microsoft Word. They will develop an understanding of which regular polygons will tessellate the plane and which will not. By completing a series of tessellations students will begin to develop an understanding of possible tessellations. The directions for this section are given on a handout.

IV. To construct a semi-regular tessellation Students will construct one of eight possible semi-regular tessellations using any method they choose, pen and ink, cut and paste or computer graphics. Students will complete a table that utilizes the concepts of the number of degrees for any angle in a regular polygon. Students will develop a conjecture for those polygons that will tessellate the plane. This conjecture they find should be similar to: The angle measure one angle of the regular must divide evenly into 360 degrees.

Materials:
  • Computer with Microsoft Word and the drawing toolbar
  • Computer with Paint on Windows 95
  • Printer
  • Handout with directions to construct a tessellation using Paint.
  • Handouts with directions on how to construct regular tessellations
  • Handout: Number of Degrees of each Angle of Regular polygons
  • Handout from Tessellation: The Geometry of Patterns by Stanley Bezuska, Margaret Kenney and Linda Silvey
  • Handouts on symmetry design from Active Learning Series Grades 6-8 Pattern Blocks by Lynda Bowen and Grant Jones
  • Colored pencils/Dot paper
  • Pattern Blocks

    Student Feedback and Assessment
    Students are assessed in several ways. A test/quiz is used to find the level of understanding for Part I of the unit.
    For Part II, the assessment is two-fold: did the students follow the directions to construct the template of the tessellation and did they use this template to cover the plane. A point system can be used to give a numeric grade for this section; additional points may be added or incorporated for originality and/or creativity of design.

    The objective for Part III is to use regular polygons to cover the plane, so the assessment should be whether or not the students were able to do so for the polygons that will tessellate. In addition, students should be able to write a short paragraph describing those polygons that do tessellate and those that do not. In this paragraph, they should include in their explanation what prevents all regular polygons from tessellating the plane.

    This explanation is not necessarily mathematically based, but more intuitive because at this point students should be exploring this concept of tessellation. In Part IV, a rubric is used to incorporate all facets of the project. This part is a project completed over a two week period outside of the classroom.

    Additional Comments and Suggestions
    This applicant included Part I of this unit as a reference. This part may be completed either before or after Parts II and III. However, it is an important section in the overall unit. Students will gain better understanding of tessellations if they are exposed to transformations, because the two concepts are closely related. Most Geometry, and many general mathematics texts, include a chapter on transformations.

    This applicant used the text: Mathematical Connections. The use of pattern blocks in this section allows students to use their tactile senses to develop an understanding of placement and arrangement of polygons in a symmetric arrangement. If Lynda Bowen's Active Learning Series on Pattern Blocks is not available, students can practice reflection by creating a design on one side of a line and then creating its mirror image on the other. By using intersecting lines and reflecting a design over these lines, students can explore the properties of rotation.

    The applicant used the Internet as a research tool to gain information on tessellations and constructing tessellations. The site that was the most helpful is Ask Dr. Math at Swarthmore University. The address is http://forum@swarthmore.edu.

    Resources

    Bezuszka, S., Kenney, M., Silvey, L. (1977). Tessellations: The Geometry of Patterns. Sunnyvale, CA: . Creative Publications.
    Bowen, L., Jones, G. (1989). Active Learning Series Grades 6-8 Pattern Blocks. Ontario: Exclusive Educational Products.
    Gardella,F.J., Fraze, P.R., Meldon, J.E., Weingarden M.S., Campbell, C. (1994). Mathematical Connections: A Bridge to Algebra and Geometry. Boston, MA: Houghton Mifflin Company.

    • Semi Regular Tessellation Project
    For this project you will construct a semi regular tessellation using one of the eight possible semi-regular tessellations. You may use a computer, a template or dot paper to help you draw your tessellation. Color in your design to emphasize a pattern. You will also complete the table of the number of degrees for an angle in a regular polygon.

    Lastly, you will write a paragraph that explains why some regular polygons tessellate the plane and some do not. Include what you have learned when completing the table and when constructing tessellations in your explanation.

    Additional Materials