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18.2

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Geometry toolkits

Activity Narrative

This optional activity provides students an opportunity to apply their learning about rigid transformations and congruent figures by creating a tessellation.

A tessellation of the plane is a regular repeating pattern of one or more shapes that covers the entire plane. Some of the most familiar examples of tessellations are seen in bathroom and kitchen tiles. Tiles (for flooring, ceiling, bathrooms, kitchens) are often composed of copies of the same shapes because they need to fit together and extend in a regular pattern to cover a large surface.

Launch

Share with students a definition of tessellation: “A tessellation is a repeating pattern of one or more shapes. The sides of the shapes fit together perfectly and do not overlap. The pattern goes on forever in all directions.” Consider showing several examples of tessellations. A true tessellation covers the entire plane. While this is impossible to show, we can identify a pattern that keeps going forever in all directions. This is important when we think about tessellations and symmetry. One definition of symmetry is, “You can pick it up and put it down a different way and it looks exactly the same.” In a tessellation, you can perform a translation and the image looks exactly the same. In the example of this tiling, the translation that takes point to point results in a figure that looks exactly the same as the one you started with. So does the translation that takes to . Describing one of these translations shows that this figure has translational symmetry.

<p>Tessellation. </p>

Provide access to geometry toolkits. Suggest to students that if they cut out a shape, it is easy to make many copies of the shape by tracing it. Encourage students to use the shapes from the previous activity (or pattern blocks if available) and experiment putting them together. They do not need to use all of the shapes, so if students are struggling, suggest that they try using copies of a couple of the simpler shapes.

Representation: Internalize Comprehension. Use multiple examples and non-examples to emphasize that a tessellation is a pattern made of shapes that have no gaps or overlaps and has translation symmetry.
Supports accessibility for: Conceptual Processing, Attention

Activity

None

Design your own tessellation. You will need to decide which shapes you want to use and make copies. Remember that a tessellation is a repeating pattern that goes on forever to fill up the entire plane.
Find a partner and trade pictures. Describe a transformation of your partner’s picture that takes the pattern to itself. How many different transformations can you find that take the pattern to itself? Consider translations, reflections, and rotations.
If there’s time, color and decorate your tessellation.

Student Response

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Building on Student Thinking

Watch out for students who choose shapes that almost-but-don’t-quite fit together. Reiterate that the pattern has to keep going forever. Often small gaps or overlaps become more obvious when the pattern continues.

Activity Synthesis

Invite students to share their designs and also describe a transformation that takes the design to itself. Consider displaying their tessellations around the room.

18.3

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Geometry toolkits

Activity Narrative

This optional activity provides students an opportunity to apply their learning about rigid transformations and congruent figures by creating designs with rotational symmetry.

They then share designs and find the different rotations (and possibly reflections) that make the shape match up with itself.

Launch

<p>Rotational symmetry figure.</p>

Ask students what transformation they could perform on the figure so that it matches up with its original position. There are a number of rotations using as the center that would work: 72 degrees or any multiple of 72 degrees. Make sure students understand that the 5 triangles in this pattern are congruent and that : This is why multiples of 72 degrees with center match this figure up with itself. They need to be careful in selecting angles for the shapes in their pattern. If they struggle, consider asking them to use pattern tiles or copies of the shapes from the previous activity to help build a pattern.

If possible, show students several examples of figures that have rotational symmetry.

Provide access to geometry toolkits. If possible, provide access to square graph paper or isometric graph paper.

Activity

None

Make a design with rotational symmetry.
Find a partner who has also made a design. Exchange designs and find a transformation of your partner’s design that takes it to itself. Consider rotations, reflections, and translations.
If there’s time, color and decorate your design.

Student Response

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Building on Student Thinking

Before now, students may think that reflection symmetry is the only kind of symmetry. Because of this, they may create a design that has reflection symmetry but not rotational symmetry. Steer students in the right direction by asking them to perform a rotation that takes the figure to itself. Acknowledge that reflection symmetry is a type of symmetry, but the task here is to create a design with rotational symmetry.

Activity Synthesis

Invite students to share their designs and also describe a transformation that takes the design to itself. Consider displaying their designs around the room.

MLR8 Discussion Supports. For each design and transformation that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Advances: Listening, Speaking