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.
Activity
None
Student Task Statement
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.
Student Response
Loading...
Building on Student Thinking
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?”
Activity Synthesis
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 , a full circle. The sum of the angles in any triangle is , which is half of 360.)
3.2
Activity
15 mins
Quadrilateral Tessellations
Standards Alignment
Building On
Addressing
8.G.A
Understand congruence and similarity using physical models, transparencies, or geometry software.
The previous activity showed how to make a tessellation with copies of a triangle. A natural question is whether or not it is possible to tessellate the plane with copies of a single quadrilateral. Students have already investigated this question for some special quadrilaterals (squares, rhombuses, regular trapezoids), but what about for an arbitrary quadrilateral? This activity gives a positive answer to this question. As students discuss this question, they develop arguments and critique each other’s reasoning (MP3). Pentagons are then investigated in the next activity, and there students will find that some pentagons can tessellate the plane while others can not.
In order to show that the plane can be tessellated with copies of a quadrilateral, students will experiment with rigid motions and copies of a quadrilateral.
This activity can be made more open ended by presenting students with a polygon and asking them if it is possible to tessellate the plane with copies of the polygon.
In the digital version of the activity, students use an applet to tessellate quadrilaterals. The applet allows students to work with many copies of each quadrilateral without tracing. The digital version may be preferable if time is limited.
Activity Synthesis
Invite some students to share their tessellations.
Discussion questions include:
“Were you able to tessellate the plane with copies of the trapezoid?” (Yes, 2 of them can be put together to make a parallelogram, and the plane can be tessellated with copies of this parallelogram.)
“What did you notice about the quadrilateral and the 180-degree rotations?” (They fit together with no gaps and no overlaps and leave space for 4 more quadrilaterals.)
“How do you know that there are no overlaps?” (The sum of the angles in a quadrilateral is 360 degrees. At each vertex in the tessellation, copies of the 4 angles of the quadrilateral come together.)
Revisit the question from the start of the activity, “Can any quadrilateral be used to tessellate the plane?” Invite students to share if their answer has changed and explain their reasoning.
3.3
Activity
20 mins
Pentagonal Tessellations
Standards Alignment
Building On
Addressing
8.G.A
Understand congruence and similarity using physical models, transparencies, or geometry software.
All triangles and all quadrilaterals give tessellations of the plane. For the quadrilaterals, this was complicated and depended on the fact that the sum of the angles in a quadrilateral is 360 degrees. Regular pentagons that do not tessellate the plane have been seen in earlier activities. The goal of this activity is to study some types of pentagons that do tessellate the plane. Students make use of structure when they relate the pentagons in this activity to the hexagonal tessellation of the plane, which they have seen earlier (MP7).
This activity can be made more open ended by presenting students with a polygon and asking them if it is possible to tessellate the plane with copies of the polygon.
In the digital version of the activity, students use an applet to tessellate a pentagon. The applet allows students to work with many copies of the pentagon without tracing. The digital version may be preferable if time is limited.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Activity Synthesis
Students may be successful in building a tessellation in the first question. The following questions guide them through a method while also asking for mathematical justification. Students who are successful in the first question can verify that their tessellation uses the strategy indicated in the following questions, and they will still need to answer the last two questions.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their explanation that the three pentagons make a hexagon. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say. . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, invite some students to share their tessellations.
Some questions to discuss include:
“Does the hexagon made by 3 copies of the pentagon tessellate the plane?” (Yes.)
“How do you know?” (I checked experimentally, or I noticed that all of the angles in the hexagon are 120 degrees.)
“Why was it important that the 2 sides of the pentagons making the 120-degree angles are congruent?” (so that when I rotate my pentagon, those 2 sides match up with each other perfectly)
“What is special about this pentagon?” (two sides are congruent, 3 angles measure 120 degrees)
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Begin the activity with, “Any triangle can be used to tile the plane (some of them in many ways). Do some quadrilaterals tessellate the plane?” (Yes: squares, rectangles, rhombuses, and parallelograms.) Next, ask, “Can any quadrilateral be used to tessellate the plane?” Give students a moment to ponder, and then poll the class for the number of yes and no responses. Record the responses for all to see. This question will be revisited in the Activity Synthesis.
Provide access to tracing paper.
Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to tracing paper, paper cut outs of the shapes, or other physical manipulatives for students to use while solving the problems. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
Student Task Statement
Can you make a tessellation of the plane with copies of the trapezoid? Explain.
Choose and trace a copy of one of the other two quadrilaterals. Next, trace images of the quadrilateral rotated 180 degrees around the midpoint of each side. What do you notice?
Can you make a tessellation of the plane with copies of the quadrilateral from the earlier problem? Explain your reasoning.
Student Response
Loading...
Building on Student Thinking
If students have trouble determining the pattern because their figures are not traced accurately, consider asking:
“What do you know about the angles in a quadrilateral?”
“How does your knowledge of quadrilateral angles connect to trying to tessellate a quadrilateral?”
Launch
Ask students:
“Can you tessellate the plane with regular pentagons?” (No.)
“Can you think of a type of pentagon that could be used to tessellate the plane?” (A square base with a 45-45-90 triangle on top, for example.)
Arrange students in groups of 2. Provide access to tracing paper.
Activity
None
Student Task Statement
Can you tessellate the plane with copies of this pentagon? Explain why or why not.
Note that the two sides making angle are congruent.
Pause your work here.
Take the pentagon and rotate it 120 degrees clockwise about the vertex at angle A, and trace the new pentagon. Next, rotate the pentagon 240 degrees clockwise about the vertex at angle A, and trace the new pentagon.
Explain why the three pentagons make a full circle at the central vertex.
Explain why the shape that the three pentagons make is a hexagon (that is, the sides that look like they are straight really are straight).
Student Response
Loading...
Building on Student Thinking
If students struggle tracing the rotated hexagon, consider asking:
“What do you already know about this hexagon?”
“What happens to the segments making angle when the hexagon is rotated about by 120 (or 240) degrees?”
