Regular Tessellations
15 min
Teacher Prep
Setup
Required Preparation
Narrative
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.
In the digital version of the activity, students use an applet to decide if regular polygons create tessellations. The applet allows students to work with many copies of each polygon without tracing. The digital version may be preferable if time is limited.
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. Students may need a reminder that regular polygons are polygons with all congruent sides and angles. 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 that could be used:
| regular polygon | tessellate? | notes |
|---|---|---|
| octagon | ||
| hexagon | ||
| pentagon | ||
| square | ||
| triangle |
Introduce the idea of a regular tessellation:
- Only one type and size of regular polygon is 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 (for example, several different polygons are 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):
Make tracing paper available to all students. Tell students that they can use the tracing paper to put together several copies of the polygons.
Student Task
-
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.
- For the polygons that do not work, what goes wrong? Explain your reasoning.
Sample Response
- The triangles, squares, and hexagons all tessellate the plane. Three hexagons meet at each vertex, four squares meet at each vertex, and six triangles meet at each vertex.
- For the pentagons, three can fit together at a vertex. There is a little extra space left, but it’s not enough to fit another pentagon. Two octagons can fit together at a vertex with space left over, but cannot tessellate the plane.
Activity Synthesis (Teacher Notes)
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 1 vertex and checked to see if there was extra space left over.)
- “Why does the pentagon not work to tessellate the plane?” (Three fit together at one vertex, but there is extra space, not enough for a fourth.)
- “Why does the octagon not work?” (Two 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.
Anticipated Misconceptions
If students are not sure how to explain their reasoning for why some of the shapes do not tessellate, consider asking:
- “Tell me more about how you tried to tessellate the shape.”
- “What is the same and what is different about this shape and one that does tessellate?”
Standards
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Regular Tessellations
15 min
Teacher Prep
Setup
Required Preparation
Narrative
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.
In the digital version of the activity, students use an applet to decide if regular polygons create tessellations. The applet allows students to work with many copies of each polygon without tracing. The digital version may be preferable if time is limited.
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. Students may need a reminder that regular polygons are polygons with all congruent sides and angles. 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 that could be used:
| regular polygon | tessellate? | notes |
|---|---|---|
| octagon | ||
| hexagon | ||
| pentagon | ||
| square | ||
| triangle |
Introduce the idea of a regular tessellation:
- Only one type and size of regular polygon is 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 (for example, several different polygons are 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):
Make tracing paper available to all students. Tell students that they can use the tracing paper to put together several copies of the polygons.
Student Task
-
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.
- For the polygons that do not work, what goes wrong? Explain your reasoning.
Sample Response
- The triangles, squares, and hexagons all tessellate the plane. Three hexagons meet at each vertex, four squares meet at each vertex, and six triangles meet at each vertex.
- For the pentagons, three can fit together at a vertex. There is a little extra space left, but it’s not enough to fit another pentagon. Two octagons can fit together at a vertex with space left over, but cannot tessellate the plane.
Activity Synthesis (Teacher Notes)
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 1 vertex and checked to see if there was extra space left over.)
- “Why does the pentagon not work to tessellate the plane?” (Three fit together at one vertex, but there is extra space, not enough for a fourth.)
- “Why does the octagon not work?” (Two 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.
Anticipated Misconceptions
If students are not sure how to explain their reasoning for why some of the shapes do not tessellate, consider asking:
- “Tell me more about how you tried to tessellate the shape.”
- “What is the same and what is different about this shape and one that does tessellate?”
Standards
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.