The goal of this task is to introduce the notion of a tessellation and then carefully examine how to create a tessellation with each pattern block shape (square, rhombus, equilateral triangle, isosceles trapezoid). For hexagons, there is only one way to fit them together because there will be gaps unless three hexagons meet at each vertex. The other shapes offer much more flexibility, and students have an opportunity to use their artistic creativity.

Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.

In the digital version of the activity, students use an applet to create tessellations. The applet allows students to select and move shapes. Use the digital version if time or physical materials are limited.

Important questions to address include:

• “Were you able to make different tessellations with your shape?” (It depends on the shape.)
• “If not, why not?” (Three hexagons have to come together at each vertex: once the first hexagon of the pattern is placed, everything else has no flexibility.)
• “If so how?” (Triangles can be built into hexagons, or they can make rows that can be translated and stacked on top of one another. Parallelograms can also be made into hexagons or rows. Trapezoids can be made into hexagons or rows. Rhombuses can be made into rows and translated. Squares can also be made into rows and translated.)
• “What does it look like to not define a tessellation?” (Two octagons can be put together sharing a vertex, but there is a gap that is not large enough for a third octagon.)

Make sure, as students share their ideas for tessellations, to use the language of rigid motions to describe the tessellations. For example, if a student has built a tessellation with parallelograms, choose two parallelograms and ask:

• “How can I use rigid motions to move one of these parallelograms into the position of the other?”
• “Are there other ways I could do this?”

In this partner activity, students take turns describing a tessellation of the plane. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). While students do not need to use the words “translation,” “rotation,” or “reflection,” their understanding of rigid motions of the plane will play a key role in explaining (and interpreting) where to place each shape in a tessellation. Communicating a geometric pattern clearly in words fully engages students in attending to precision (MP6).

For the Warm-up, each student will describe a tessellation while the partner identifies a picture of the tessellation. After a brief discussion of what language was most helpful, students then take turns describing a tessellation as their partner attempts to build the tessellation. They switch roles and then reflect on any misinterpretations that happened and how they may have been related to the language used to describe the tessellations.

After students recreate their partner's tessellation, discuss as a whole class the use of the words “translate,” “reflect,” and “rotate” (or equivalents).

• “When you used a translation, did you specify the direction and how far?” (No, for Card B I told my partner to start with a square and then to draw a congruent square so they would share half a side.)
• “When a rotation was involved, did you specify the number of degrees of the rotation?” (Yes, for the rows of equilateral triangles on Card A, I said to rotate 180 degrees about the midpoint of the side of the triangle to draw the next triangle.)
• “When a rotation was involved, did you specify the center of rotation?” (Answers vary)
• “Did you use any reflections?” (Answers vary)