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