What’s Regular About Tesselations?

Bibliographic Information:
Miles, Victoria. “What’s regular about Tessellations?” Mathematics Teaching in the Middle School, NCTM, Feb 2000.

Mathematical Concept: Regular Tiling

Grade Levels: 6th – 8th

NC Standard Course of Study:
Competency Goal 3: The learner will understand and use properties and relationships in geometry.
3.01: Represent problem situations with geometric models.
3.03: Identify, predict, and describe dilations in the coordinate plans.

Materials:
Scissors (optional)
Glue (optional)
Colorful Poster Paper (optional)
Snack Size Zip-Lock Bags (1 per student)
Computer with Internet Access
Overhead: What’s regular about this Polygon?
Handout: What’s regular about Tessellations (activity sheet)
Handout: Multiple copies of “Regular Polygons Activity Sheet” per student on (optional – colorful paper)
Answer Key: What’s Regular about Tessellations?

Detailed Description:
In this lesson, students explore regular and semi-regular tessellations. Students use manipulatives to discover which regular polygons will tessellate and which will not. Students will use geometry and measurement to investigate the three regular and eight semi-regular tessellations. This lesson works best when used with the Tessellation Creator. However, if computers with Internet connections are not available, you may use the Regular Polygons Activity sheet in its place. Have several copies available for each student so they are not limited in their explorations by having too few shapes. Each student will also need scissors, glue, and poster paper if using the cut-outs.

Prior to presenting this lesson, the teacher should become familiar with regular and semi-regular tessellations prior to teaching the lesson. There are 3 regular and 8 semi-regular tessellations (as shown on the “What’s Regular About Tessellations?” answer key. The teacher will want to explore how to create the various tessellations using the Tessellation Creator manipulative website.

Procedure:
1. If a computer with Internet access is not available, hand out the “Regular Polygons” activity sheet, scissors, and poster paper; have students multiple copies of this handout so that their exploration isn’t limited by having too few shapes.

2. Pass out copies of the “What’s Regular About This Polygon?” handout and/or use an overhead projector, to introduce the lesson.
a. Allow students time to answer the questions in class.
b. Review the definition of a regular polygon.
c. Introduce/Review methods for calculating the measure of one interior angle of a regular polygon.
d. There are 2 ways to show students the answer to Question 3:
i. Select a vertex and draw the two diagonals from that vertex. The pentagon is now divided into 3 triangles, each has 180°. Therefore, the sum of the interior angle measure of the pentagon is 180 x 3 = 540°.
ii. Draw a point inside the pentagon, and draw 5 line segments connecting that point to the 5 vertices. Notice the pentagon has been divided into 5 triangles. This time 180° x 5 = 900°, but 360° must be subtracted away from the 900° since the angles inside the pentagon don’t pertain to the pentagon’s interior angle.

3. Distribute one “What’s Regular About Tessellations?” activity sheet to each student. (Optional: at this time consider allowing students to work with partners so they can assist each other when using the technology tools and thinking about angle measures.)
a. Question 1 on this worksheet should be a review for students. They should already be familiar with the sum of the interior angles of polygons before beginning this lesson.
b. Students will discover that there are 3 regular tessellations: equilateral triangles, squares, and regular hexagons.
c. The reason these polygons tessellate on their own is the measure of a single interior angle in each polygon is a factor of 360°.
d. Stress connection between interior angle measures and tessellations.
e. If the teacher feels as if the students need more review, consider having them explore the Angle Sums manipulative website before beginning the formal lesson.

4. If students complete the table for regular tessellations, let them know there are 8 semi-regular tessellations. Have students find as many semi-regular tessellations as time permits.
a. Assign groups of 4-6 students a different semi-regular tessellation to share with the class.
b. Have students tape or glue the paper polygons onto poster paper.
i. Each poster should include:
1. a tiling of their semi-regular tessellation using colorful paper polygons
2. The vertex configuration
3. The sum of the interior angles surrounding any vertex.
ii. During the brief presentations, ask students to state:
1. What regular polygons are used?
2. If there are any other ways to classify the vertex configuration
3. Why it’s a semi-regular tessellation
iii. Collect the posters to be used a class bulletin board to review tessellation. (This is not included in the original lesson plan from www.NCTM.org )

Attachment: Worksheet/Activity Student Page
Original “What’s Regular About Tessellations?’ lesson from NCTM.org (for accommodation suggestions,
questions for students, and extension assignments)
Handout/Overhead: What’s Regular About This Polygon?
Activity Sheet: What’s Regular About Tessellations?
Activity Sheet (optional): Regular Polygons
Answer Key: What’s Regular about Tessellations?

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