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Regular Tessellations
by Suzanne Alejandre
A Math Forum Companion Lesson to:
NOTE: Use of Internet Explorer 5.0 is recommended.
Getting started: Go to BoxerMath's Tessellation Tool, which will allow you to build tessellations and other designs by attaching the corners of various shapes. Note: BoxerMath's Tessellation Tool will open in a second window. Take a moment to adjust the two windows so that you can work with the applet and also view the text of the activity.
Challenge: Can you use triangle(s) and the translate feature and/or rotate feature to cover the plane with no gaps or overlaps?
Manipulatives: Use the triangles in a set of activity pattern blocks to make a tessellation. (Paper activity pattern blocks are available on the Web; see Hand Made Manipulative Instructions by Margo Mankus.) Revisiting the activity: Now that you have tessellated with BoxerMath's Tessellation Tool and with activity pattern blocks, return to the Tessellation Tool to think about using only one equilateral triangle.
Formalizing the mathematics:
Focus on either of the two units:
Before returning to BoxerMath's Tessellation Tool, predict whether a square will tessellate by answering these questions: Repeat the process with a hexagon. Again make a prediction by asking yourself: Now that you have considered the cases of the equilateral triangle, the square, and the regular hexagon, can you complete the following chart?
Can you fill in the blanks in the following statement? In a tessellation the polygons used will fit together with their angles arranged around a point with no gaps or overlaps. When using just one polygon (for example, only equilateral triangles), the interior measure of each angle will need to be a factor of _____ degrees (meaning that ____ degrees can be divided evenly by that angle measure). The only regular polygons that qualify are the __________________, ___________________, and ___________________. Assessment:

Math Forum Resources
Information and Lessons on tessellations and symmetry:
[Math Forum Resources] Selections from Math Forum Problems of the Week:
[Math Forum Resources] Selections from Ask Dr. Math Archives:

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BoxerMath.com offers online curricula for Prealgebra, Algebra I & II, Geometry, and Trigonometry. All courses include practice problems, tests, lesson plans and enrichment activities. BoxerMath.com also correlates course lessons with state and national mathematics standards. 