Grade Level: 10 - 12

Subject(s): Language Arts, Mathematics, Art

Learner Outcomes:

The student will be able to determine the characteristics of the regular polygons that will tessellate.

The student will be able to determine if a non-regular polygon will tessellate.

• Files: square.jpg, triangle.jpg, pentagon.jpg, hexagon.jpg, octagon.jpg
• HTML format of the Discover worksheet. Microsoft Word format of the Discover Worksheet (Zipped).

• Computer
• PC-Paint or other drawing software

• The students should know how to find the sum of the angles in a polygon.
• The students should know how to find the angles in a regular polygon.
• Before the students work on the Discover Worksheet, you will need to place the picture files on the students' disk.

Depending on the students' ability in using PC-Paint, copying, pasting, and manipulating images, this lesson could take one to two 50 minute class periods.

1. The students follow the procedures in the Discover Worksheet.
2. The students are instructed to save the files in the form as "flast#."  For example, if the student's name is John Smith and he is saving the first image, the file name would be "jsmith1."
3. Help the students discover the following:
• each angle of the equilateral triangle is 60^o.
• each angle of the square is 90^o.
• each angle of the regular hexagon is 120^o.
• 360^o is divisible by 60^o, 90^o, and 120^o.
• each angle of the regular pentagon is 108^o.
• each angle of the regular octagon is 135^o.
• 360^o is not divisible by 108^o, or 135^o.
• Since the angles of a regular heptagon, nonagon, and decagon are 128.6^o, 140^o, and 144^o respectively, none of these polygons will tessellate.
• The theorem the students should discover is that the only regular polygons that will tessellate are the equilateral triangle, square, and regular hexagon.
• The proof of the theorem is illustrated at the right.
1. All of the interior angles of a regular polygon are equal.
2. In order to tessellate, the angles must come together at one vertex.
3. The sum of the angles must be 360^o.
4. The interior angle of the polygon must be a divisor of 360^o.

For students who have difficulty using the computer or PC-Paint, the discovery activity can be accomplished using "Pattern Blocks."  The students should have six each of the regular polygons (equilateral triangle, square, regular pentagon, regular hexagon, and regular octagon). As an alternative to the pattern blocks, cutouts of the shapes could be used.  Use the patterns in the polygon files as a template.

The procedure above involved regular polygons that tessellate.  Mixing polygons, or using non-regular polygons, other shapes can also tessellate.  The Irregular Tessellations worksheet will give the students an opportunity to explore other types of tessellations. Click here for a Microsoft Word form of the Irregular Tessellations worksheet.

IGO's:

English Language Arts: 10.24
Geometry:  G.8, G.14, G.19, G.23
Instructional Practices: IP.6, IP.7, IP.12,  IP.20

References:

• Tessellations, Creative combinations of Math, Art and fun
• Tessellations Tutorials
• Totally Tessellated

Created by:

Rusty Campbell
David Underwood
                            Carol White

North Marion High School
Rt. 1, Box 100
Farmington, WV  26571

Date modified:

Attachments:

• HTML version of the Discover Worksheet
• Microsoft Word version of the Discover Worksheet (Zipped)
• HTML version of Irregular Tessellations
• Microsoft Word version of Irregular Tessellations