Tessellating Polygons
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this activity, students experiment with copies of a triangle (no longer equilateral) and discover that it is always possible to build a tessellation of the plane. A key in finding a tessellation with copies of a triangle is to experiment with organizing copies of the triangle, and then reasoning that two copies of a triangle can always be arranged to form a parallelogram. Students may not remember this construction from the sixth grade, but with copies of the triangle to experiment with, they will find the parallelogram or a different method. These parallelograms can then be put together in an infinite row, and these rows can then be stacked upon one another to tessellate the plane. Applying this process to a variety of triangles gives the opportunity to apply repeated reasoning to develop the concept that all triangles can tessellate the plane (MP8).
In the digital version of the activity, students use an applet to tessellate the triangles. The applet allows students to work with many copies of each triangle without tracing. The digital version may be preferable if time is limited.
Launch
Assign different triangles to different students or groups of students. Provide access to tracing paper.
If students finish early, consider asking them to work on building a different tessellation or coloring their tessellation.
Student Task
Your teacher will assign you one of the three triangles. You can use the picture to draw copies of the triangle on tracing paper. Your goal is to find a tessellation of the plane with copies of the triangle.
Sample Response
Sample response for each type of triangle:
Activity Synthesis (Teacher Notes)
Invite several students to share their tessellations for all to see.
Consider asking the following questions to help summarize the lesson:
- “Were you able to make a tessellation with copies of your triangle?” (Most students should respond “yes.”)
- “How did you know that you could continue your pattern indefinitely to make a tessellation?” (All parallelograms can be used to tessellate the plane as they can be placed side by side to make infinite “rows” or “columns,” and then these rows or columns can be displaced to fill up the plane.)
Share some of the tessellation ideas students come up with and relate them back to previous work, that is the tessellation of the plane with rectangles and parallelograms.
If time allows, ask:
- Will all triangles tessellate? How do you know? (Yes, any triangle will tessellate because any triangle can be rotated with itself to form a parallelogram that will tessellate.)
- What do you notice about the angles of the triangles that are in a tessellation? (The angles that meet at each vertex must add to 360∘, a full circle. The sum of the angles in any triangle is 180∘, which is half of 360.)
Anticipated Misconceptions
If students struggle to put together copies of their triangle in a way that can be continued to tessellate the plane, consider asking:
- “Tell me more about how you have put triangles together.”
- “What are ways you can combine just two copies of the triangle?”
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
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Tessellating Polygons
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this activity, students experiment with copies of a triangle (no longer equilateral) and discover that it is always possible to build a tessellation of the plane. A key in finding a tessellation with copies of a triangle is to experiment with organizing copies of the triangle, and then reasoning that two copies of a triangle can always be arranged to form a parallelogram. Students may not remember this construction from the sixth grade, but with copies of the triangle to experiment with, they will find the parallelogram or a different method. These parallelograms can then be put together in an infinite row, and these rows can then be stacked upon one another to tessellate the plane. Applying this process to a variety of triangles gives the opportunity to apply repeated reasoning to develop the concept that all triangles can tessellate the plane (MP8).
In the digital version of the activity, students use an applet to tessellate the triangles. The applet allows students to work with many copies of each triangle without tracing. The digital version may be preferable if time is limited.
Launch
Assign different triangles to different students or groups of students. Provide access to tracing paper.
If students finish early, consider asking them to work on building a different tessellation or coloring their tessellation.
Student Task
Your teacher will assign you one of the three triangles. You can use the picture to draw copies of the triangle on tracing paper. Your goal is to find a tessellation of the plane with copies of the triangle.
Sample Response
Sample response for each type of triangle:
Activity Synthesis (Teacher Notes)
Invite several students to share their tessellations for all to see.
Consider asking the following questions to help summarize the lesson:
- “Were you able to make a tessellation with copies of your triangle?” (Most students should respond “yes.”)
- “How did you know that you could continue your pattern indefinitely to make a tessellation?” (All parallelograms can be used to tessellate the plane as they can be placed side by side to make infinite “rows” or “columns,” and then these rows or columns can be displaced to fill up the plane.)
Share some of the tessellation ideas students come up with and relate them back to previous work, that is the tessellation of the plane with rectangles and parallelograms.
If time allows, ask:
- Will all triangles tessellate? How do you know? (Yes, any triangle will tessellate because any triangle can be rotated with itself to form a parallelogram that will tessellate.)
- What do you notice about the angles of the triangles that are in a tessellation? (The angles that meet at each vertex must add to 360∘, a full circle. The sum of the angles in any triangle is 180∘, which is half of 360.)
Anticipated Misconceptions
If students struggle to put together copies of their triangle in a way that can be continued to tessellate the plane, consider asking:
- “Tell me more about how you have put triangles together.”
- “What are ways you can combine just two copies of the triangle?”
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.