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.

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.

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.

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)