# Lesson 2

Regular Tessellations

## 2.1: Regular Tessellations (15 minutes)

### Optional activity

The goal of this activity is to introduce a regular tessellation of the plane and conjecture which shapes give regular tessellations. Students construct arguments for which shapes can and cannot be used to make a regular tessellation (MP3). The focus is on experimenting with shapes and noticing that in order for a shape to make a regular tessellation, we need to be able to put a whole number of those shapes together at a single vertex with no gaps and no overlaps. This greatly limits what angles the polygons can have and, as a result, there are only three regular tessellations of the plane. This conjecture will be demonstrated in the other two activities of this lesson.

### Launch

Display a table for all to see with at least two columns keeping track of which regular polygons make a tessellation and which do not. A third column could be used for extra comments (for example, about angle size of the polygon or other remarks). Here is an example of a table you might use:

shape tessellate? notes
octagon
hexagon
pentagon
square
triangle

Introduce the idea of a regular tessellation:

• Only one type and size of polygon used.
• If polygons meet, they either share a single vertex or a single side.
• Show some pictures of tessellations that are not regular, and ask students to identify why they are not (e.g., several different polygons used, edges of polygons do not match up completely). Ask students which of the tessellations pictured here are regular tessellations (only the one with squares):

For the print version, make tracing paper available to all students. Tell students that they can use the tracing paper to put together several copies of the polygons.

For the digital version tell students that they can click on the shapes to make several copies.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “The _____ can make a regular tessellation of the plane because . . .”, “I noticed _____ so I . . .”, or “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

1. For each shape (triangle, square, pentagon, hexagon, and octagon), decide if you can use that shape to make a regular tessellation of the plane. Explain your reasoning.

2. For the polygons that do not work what goes wrong? Explain your reasoning.

### Launch

Display a table for all to see with at least two columns keeping track of which regular polygons make a tessellation and which do not. A third column could be used for extra comments (for example, about angle size of the polygon or other remarks). Here is an example of a table you might use:

shape tessellate? notes
octagon
hexagon
pentagon
square
triangle

Introduce the idea of a regular tessellation:

• Only one type and size of polygon used.
• If polygons meet, they either share a single vertex or a single side.
• Show some pictures of tessellations that are not regular, and ask students to identify why they are not (e.g., several different polygons used, edges of polygons do not match up completely). Ask students which of the tessellations pictured here are regular tessellations (only the one with squares):

For the print version, make tracing paper available to all students. Tell students that they can use the tracing paper to put together several copies of the polygons.

For the digital version tell students that they can click on the shapes to make several copies.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “The _____ can make a regular tessellation of the plane because . . .” , “I noticed _____ so I . . .”, or “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

1. For each shape (triangle, square, pentagon, hexagon, and octagon), decide if you can use that shape to make a regular tessellation of the plane. Explain your reasoning.

2. For the polygons that do not work, what goes wrong? Explain your reasoning.

### Anticipated Misconceptions

If students working with the pentagon and octagon add other shapes to make a more complicated tessellation, remind them that a regular tessellation uses copies of a single shape.

### Activity Synthesis

To help students think more about what shapes do and do not tessellate and why, ask:

• “Which polygons appear to tessellate the plane?” (Square, equilateral triangle, hexagon.)
• “How did you decide?” (I put as many together as I could at one vertex and checked to see if there was extra space leftover.)
• “Why does the pentagon not work to tessellate the plane?” (3 fit together at one vertex, but there is extra space, not enough for a fourth.)
• “Why does the octagon not work?” (2 fit together, but there is not enough space for a third.)

During the discussion, fill out the table, indicating that it is possible to make a tessellation with equilateral triangles, squares, and hexagons, but not with pentagons or octagons.

## 2.2: Equilateral Triangle Tessellation (15 minutes)

### Optional activity

The goal of this activity is to verify, via angle calculations, that equilateral triangles (and hence) regular hexagons can be used to make regular tessellations of the plane. Students have encountered the equilateral triangle plane tessellations earlier in grade 8 when working on an isometric grid. In order to complete their investigation of regular tessellations of the plane, it remains to be shown that no other polygons work. This will be done in the next activity.

Students are required to reason abstractly and quantitatively (MP2) in this activity. Tracing paper indicates that six equilateral triangles can be put together sharing a single vertex. Showing that this is true for abstract equilateral triangles requires careful reasoning about angle measures.

### Launch

In the previous task, equilateral triangles, squares, and hexagons appeared to make regular tessellations of the plane. Tell students that the goal of this activity is to use geometry to verify that they do.

Refer students to regular polygons printed in the previous activity for a visual representation of an equilateral triangle.

Representation: Internalize Comprehension. Represent the same information through different modalities by using physical objects to represent abstract concepts. Some students may benefit by using virtual or concrete manipulatives (geogebra, tracing paper, or equilateral triangle cut-outs) to connect abstract concepts to concrete objects.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

1. What is the measure of each angle in an equilateral triangle? How do you know?
2. How many triangles can you fit together at one vertex? Explain why there is no space between the triangles.

3. Explain why you can continue the pattern of triangles to tessellate the plane.

4. How can you use your triangular tessellation of the plane to show that regular hexagons can be used to give a regular tessellation of the plane?

### Anticipated Misconceptions

Students may know that an equilateral triangle has 60-degree angles but may not be able to explain why. Consider prompting these students for the sum of the three angles in an equilateral triangle.

Students may not see a pattern of hexagons within the triangle tessellation. Consider asking these students what shape they get when they put 6 equilateral triangles together at a single vertex.

### Activity Synthesis

Consider asking the following questions to lead the discussion of this activity:

• “How did you find the angle measures in an equilateral triangle?” (The sum of the angles is 180 degrees, and they are all congruent so each is 60 degrees.)
• “Why is there no space between six triangles meeting at a vertex?” (The angles total 360 degrees, which is a full circle.)
• “How does your tessellation with triangles relate to hexagons?” (You can group the triangles meeting at certain vertices into hexagons, which tessellate the plane.)
• “Are there other tessellations of the plane with triangles?” (Yes. You can make infinite rows of triangles that can be placed on top of one another—and displaced relative to one another.)

Consider showing students an isometric grid, used earlier in grade 8 for experimenting with transformations, and ask them how this relates to tessellations. (It shows a tessellation with equilateral triangles.)

Point out that this activity provides a mathematical justification for the “yes” in the table for triangles and hexagons.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After a response to one of the discussion questions is shared with the class, invite students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. This will provide more students with an opportunity to practice using language about tessellations.
Design Principle(s): Support sense-making

## 2.3: Regular Tessellation for Other Polygons (15 minutes)

### Optional activity

The goal of this activity is to show that only triangles, squares, and hexagons give regular tessellations of the plane. The method used is experimentation with other regular polygons. The key observation is that the angles on regular polygons get larger as we add more sides, which is a good example of observing structure (MP7). Since three is the smallest number of polygons that can meet at a vertex in a regular tessellation, this means that once we pass six sides (hexagons), we will not find any further regular tessellations. The activities in this lesson now show that there are three and only three regular tessellations of the plane: triangles, squares, and hexagons.

### Launch

Ask students “Are there some other regular polygons, in addition to equilateral triangles, squares, and hexagons, that can be used to give regular tessellations of the plane?” Some students may suggest regular polygons with more sides than the ones they have seen already, others may think that there are no other possibilities. Tell students that for this activity, they are going to investigate polygons with 7, 8, 9, 10, and 11 sides to see if they do or do not tessellate and why.

Print version: Provide access to tracing paper and protractors and tell students that they can use these to explore their conjectures.

Digital version: Tell students to use the app to explore their conjectures.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

1. Can you make a regular tessellation of the plane using regular polygons with 7 sides? What about 9 sides? 10 sides? 11 sides? 12 sides? Explain.

2. How does the measure of each angle in a square compare to the measure of each angle in an equilateral triangle? How does the measure of each angle in a regular 8-sided polygon compare to the measure of each angle in a regular 7-sided polygon?

3. What happens to the angles in a regular polygon as you add more sides?

4. Which polygons can be used to make regular tessellations of the plane?

### Launch

Ask students “Are there some other regular polygons, in addition to equilateral triangles, squares, and hexagons, that can be used to give regular tessellations of the plane?” Some students may suggest regular polygons with more sides than the ones they have seen already, others may think that there are no other possibilities. Tell students that for this activity, they are going to investigate polygons with 7, 8, 9, 10, and 11 sides to see if they do or do not tessellate and why.

Print version: Provide access to tracing paper and protractors and tell students that they can use these to explore their conjectures.

Digital version: Tell students to use the app to explore their conjectures.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

1. Can you make a regular tessellation of the plane using regular polygons with 7 sides? What about 9 sides? 10 sides? 11 sides? 12 sides? Explain.

2. How does the measure of each angle in a square compare to the measure of each angle in an equilateral triangle? How does the measure of each angle in a regular 8-sided polygon compare to the measure of each angle in a regular 7-sided polygon?
3. What happens to the angles in a regular polygon as you add more sides?

4. Which polygons can be used to make regular tessellations of the plane?