## What’s Regular About Tessellations?

- Lesson

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.

### Preparing for the Lesson

It may be helpful for you to be familiar with the regular and semi-regular tessellations prior to teaching this lesson. There are 3 regular and 8 semi-regular tessellations, as shown on the What’s Regular About Tessellations? Answer Key. Explore how to create the various tessellations using the Tessellation Creator.

What’s Regular About Tessellations? Answer Key

This lesson works best when used with the Tessellation Creator. However, if computers (or tablets) 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.

Regular Polygons Activity Sheet

Question 1 on the What’s Regular About Tessellations? Activity Sheet should be a review.

What’s Regular About Tessellations? Activity Sheet

Students should already be familiar with the sum of the interior angles of polygons before beginning this lesson. If you feel your students need a review, consider having them explore the Angle Sums Tool before beginning the formal lesson.

### Introducing the Lesson

Begin the lesson by showing the What’s Regular About This Polygon? Overhead.

What’s Regular About This Polygon? Overhead

Allow students time to answer the questions on Page 1, then review the answers on Page 2. Use whole-class instruction to review the definition of a regular polygon. Introduce or review methods for calculating the measure of one interior angle of a regular pentagon.

There are a couple of ways to show students the answer to Question 3:

- Select a vertex and draw the two diagonals from that vertex. The pentagon is now divided into three triangles, each of which has 180°. Therefore, the sum of the interior angle measures of the pentagon is 180 × 3 = 540°.
- 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° × 5 = 900°, but 360° must be subtracted away from 900° since the angles inside the pentagon don’t pertain to the pentagon’s interior angle measures.

### Student Exploration

Distribute one What’s Regular About Tessellations? Activity Sheet to each student.

What’s Regular About Tessellations? Activity Sheet

Consider allowing students to work with
partners, so they can assist each other when using the technology tools
and thinking about angle measures. Consider using the accommodations
suggested at the end of the *Instructional Plan*.

As students explore the Tessellation Creator they will discover there are 3 regular tessellations: regular triangles, regular quadrilaterals (squares), and regular hexagons.

The reason these polygons tessellate on their own is the measure of a single interior angle in each polygon is a factor of 360°. You may need to address individual students or the whole class to stress the connection between interior angle measures and tessellations.

Although there is information about semi-regular tessellations included on the top of Page 2 activity sheet, it is important to check for student understanding throughout the lesson. Consider asking students to explain the vertex configuration of a tessellation they recorded on their activity sheets. Some questions you might ask them are:

- How do you know your tessellation will tile the plane?
- What is the sum of the interior angle measures of polygons surrounding a vertex?

If students complete the table for semi-regular tessellations, let them know there are 8 semi-regular tessellations. Have students find as many semi-regular tessellations as time permits.

To initiate a class summary, assign groups of 4–6 students a different semi-regular tessellations to share with the whole class during the summary time. Photocopying polygon templates onto colorful paper will make it easier for students to recreate their tessellations. Have students tape or glue the paper polygons onto poster paper. Each poster should include:

- a tiling of their semi-regular tessellation using colorful paper polygons
- the vertex configuration
- the sum of the interior angles surrounding any vertex

During the summary, ask students questions about their tessellations such as:

- What regular polygons are used in your tessellation?
- Is there any other way to classify the vertex configuration?
- Why is it a semi-regular tessellation?

### Accommodations

Consider providing 1 or 2 vertex configurations to students with weak spatial reasoning. Students can use the vertex configuration information to construct tessellations. For example, if you give a student {12,12,3}, students will surround a point using 2 dodecagons and 1 triangle; then use the vertex configuration to build off that set and create a tessellation.

Consider modifying the set of polygons to include only triangles, squares, and hexagons. This will allow students to discover 4 of the 8 semi-regular tessellations.

Give students with learning differences a partially or fully completed table of interior angle measures of polygons (Question 1) .

For students with fine motor challenges, have the polygons cut out so they need only glue them down onto the poster paper for the class summary.

### Reference

Mathematics Teaching in the Middle School, NCTM, Feb 2000, Blake Peterson “From Tessellations to Big Polyhedra”

- Scissors (optional)
- Glue (optional)
- Poster paper (optional)
- Computers or tablets with internet access
- What’s Regular About This Polygon? Overhead
- What’s Regular About Tessellations? Activity Sheet
- Regular Polygons Activity Sheet (optional)
- What’s Regular About Tessellations? Answer Key

**Assessment Options**

- Have students complete an exit slip by answering 3 questions, such as these:
- Define regular tessellation.
- Describe a vertex configuration of {3,4,6,4}. What does each number represent?
- True or false: A dodecagon, 2 triangles and 1 square will surround a point completely. Explain.

- Show an image of a regular and a semi-regular tessellation. Ask students to classify each tessellation and give the vertex configuration of each.

**Extensions**

- Have students create tilings which are not regular or semi-regular tessellations, but which still tile the plane. Explore ways to classify these tessellations in a way similar to vertex configurations.
- One of the
*Questions for Students*you may have asked is whether or not a combination of pentagons and hexagons will tile the plane. Although this is not possible, there is a three-dimensional polyhedron that is constructed using only pentagons and hexagons, the truncated icosahedron or soccer ball. Ask students to explain why a three-dimensional model can be constructed using pentagons and hexagons, but a two-dimensional model can not. - This lesson extends naturally into an introduction to regular
and semi-regular polyhedra. Allow students to extend what they learned
about tessellations to construct the 5 regular and 13 semi-regular
polyhedra. Polygon manipulatives, such as Polydrons, are useful for
this exploration. The Geometric Solids Tool is an effective introduction to three-dimensional geometry terms such as face, vertex, and edge.

Geometric Solids Tool

**Questions for Students**

1. Is it possible to tile the plane using only pentagons? Why or why not?

[It is not possible because the interior angle measure of a regular pentagon is 108°, which is not a factor of 360°.]

2. Is it possible to tile the plane using some combination of pentagons and hexagons? Explain using interior angle measures.

[It is not possible. The interior angle of a pentagon measures 108° and a hexagon measures 120°. With 1 pentagon and 2 hexagons: 108° + (2 × 120°) = 348°; with 2 pentagons and 1 hexagon: (2 × 108°) + 120° = 336°; In either case, the space left is not big enough for an additional pentago or hexagon.]

3. Why is the tessellation shown not a regular or semi-regular tessellation?

[This is a tessellation since it covers the plane. However, it is not a regular or semi-regular tessellation because there is no consistent vertex configuration. For example, one shown vertex has a configuration of {3,4,3,12} while another vertex shows {4,3,4,6}. In addition, regular and semi-regular tessellations are constructed with only regular polygons. There are non-square rhombi in this tiling, which are not regular polygons.]

**Teacher Reflection**

- Did your students have prior knowledge about interior angle measures of regular polygons? How could you pre-assess their knowledge and adapt the lesson in the future?
- What evidence did you see that working in groups helped students learn?
- What other forms of assessment could be used in conjunction with this lesson?

### Learning Objectives

Students will:

- Review the sum of the interior angle measures of a polygon.
- Use manipulatives and interior angle measures of polygons to identify three regular tessellations.
- Use manipulatives and relationships among polygons to identify eight semi-regular tessellations.
- Be able to classify tessellations using vertex configurations.

### NCTM Standards and Expectations

- Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

- Use geometric models to represent and explain numerical and algebraic relationships.

### Common Core State Standards – Mathematics

Grade 7, Geometry

- CCSS.Math.Content.7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Grade 7, Geometry

- CCSS.Math.Content.7.G.B.5

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Grade 8, Geometry

- CCSS.Math.Content.8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP7

Look for and make use of structure.

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.