Lesson 18

Throughout this lesson, students build different patterns with copies of some polygons. In this activity, they make some copies of each polygon and arrange them in a circle. They calculate some of the angles of the polygons while also gaining an intuition for how the polygons fit together. Here are the figures included in the blackline master:

Students might use a protractor to measure angles, but the measures of all angles can also be deduced. In the first question in the task, students are instructed to fit copies of an equilateral triangle around a single vertex. Six copies fit, leading them to deduce that each angle measures \(60^\circ\) because \(360 \div 6 = 60\). For the other shapes, they can reason about angles that sum to \(360^\circ\), angles that sum to a line, and angles that sum to a known angle.

Provide access to geometry toolkits. Distribute one half-sheet (that contains 7 shapes) to each student. It may be desirable to demonstrate how to use tracing paper to position and trace copies of the triangle around a single vertex, as described in the first question.

When deducing angle measures, it is important to know that angles "all the way around" a vertex sum to \(360^\circ\). It is also important to know that angles that make a line when adjacent sum to \(180^\circ\). Monitor for students who need to be reminded of these facts.

For the remainder of the lesson, it is not so important that the degree measures of the angles are known, so don’t dwell on the answers. Select a few students who deduced angles' measures by fitting pieces together to present their work. Make sure students see lots of examples of shapes fitting together like puzzle pieces.

Recall from the previous lesson that the 3 congruent angles in an equilateral triangle make a line or 180-degree angle, so it makes sense that 6 copies of this angle make a full circle.

Each classroom activity in this lesson (this one, creating a tessellation, and the next one, creating a design with rotational symmetry) could easily take an entire class period or more. Consider letting students choose to pursue one of the two activities.

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.

Share with students a definition of tessellation, like, “A tessellation of the plane is a regular repeating pattern of one or more shapes that covers the entire plane.” 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 \(Q\) to point \(R\) results in a figure that looks exactly the same as the one you started with. So does the translation that takes \(S\) to \(Q\). Describing one of these translations shows that this figure has translational symmetry.

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.

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 you try to continue the pattern.

Invite students to share their designs and also describe a transformation that takes the design to itself. Consider decorating your room with their finished products.

Each classroom activity in this lesson (the previous one, creating a tessellation, and this one, creating a design with rotational symmetry) could easily take an entire class period or more. Consider letting students choose to pursue one of the two activities.

In this activity, using their geometry toolkits, students can make their own design that has rotational symmetry. They then share designs and find the different rotations (and possibly reflections) that make the shape match up with itself.

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 \(A\) as the center that would work: \(72^\circ\) or any multiple of \(72^\circ\). Make sure students understand that the 5 triangles in this pattern are congruent and that \(5 \boldcdot 72 = 360\): This is why multiples of \(72^\circ\) with center \(A\) 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.

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.

Invite students to share their designs and also describe a transformation that takes the design to itself. Consider decorating your room with their finished products.