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Regular Tessellations
by Suzanne Alejandre
A Math Forum Companion Lesson to:
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NCTM Standards alignment:
Objective: By interacting with BoxerMath's Tessellation Tool, students understand why equilateral triangles, squares, and regular hexagons tessellate "regularly" in the Euclidean plane. Introducing the activity: BoxerMath's Tessellation Tool allows the student to build tessellations and other designs by attaching the vertices of various polygons to one another. 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. After an introductory exploration period, review the features of BoxerMath's Tessellation Tool that the students have discovered including:
Challenge: Ask the students to use triangle(s) and by translating and rotating, cover the plane with no gaps or overlaps. Here are two samples:
Using manipulatives: Students' mathematical understanding can be extended if a combination of technology and concrete manipulatives is used. Provide students with activity pattern blocks. If they work with partners or in a group, challenge them to make "different" tessellations using only the equilateral triangles. (Paper activity pattern blocks are available on the Web; see Hand Made Manipulative Instructions by Margo Mankus.) Revisiting the activity: Now that the students have tessellated with BoxerMath's Tessellation Tool and with activity pattern blocks, have them return to the Tessellation Tool to think about using only one equilateral triangle.
Formalizing the mathematics:
Focus students' attention on either of the two units:
Before returning to BoxerMath's Tessellation Tool, ask the students to predict whether a square will tessellate by answering these questions: After students have tested their prediction, repeat the process with a hexagon. Again ask them to make a prediction by asking: Now that the students have considered the cases of the equilateral triangle, square, and regular hexagon, ask them to help complete the chart: NOTE: Numbers indicated in red would not be revealed to students.
After discussing the numbers needed to complete the table, help students come to the following conclusion: 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 360 degrees (meaning that 360 degrees can be divided evenly by that angle measure). The only regular polygons that qualify are the equilateral triangle, the square, and the regular hexagon. 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 Teacher2Teacher Discussions:
[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. 