Unit 1, Lesson 18
Rotate and Tessellate
Accelerated 7
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To Gather
Geometry toolkits
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 degrees because
Provide access to geometry toolkits. Distribute one half-sheet (that contains 7 shapes) to each student. Give students 1–2 minutes of individual work time. Pause students after the first question, and invite students to share how they arranged the triangles around a single vertex. Demonstrate using tracing paper if needed. Give students an additional 3–5 minutes of work time before a whole-class discussion.
Your teacher will give you some shapes.
How many copies of the equilateral triangle can you fit together around a single vertex, so that the triangles’ edges have no gaps or overlaps? What is the measure of each angle in these triangles?
What are the measures of the angles in the
Square?
Hexagon?
Parallelogram?
Right triangle?
Octagon?
Pentagon?
When deducing angle measures, it is important to know that angles "all the way around" a vertex sum to 360 degrees. It is also important to know that angles that make a line when adjacent sum to 180 degrees. 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.
Remind students that the 3 congruent angles in an equilateral triangle make a straight angle, so it makes sense that 6 copies of this angle make a full circle.
Optional
To Gather
Geometry toolkits
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.
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
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.
Invite students to share their designs and also describe a transformation that takes the design to itself. Consider displaying their tessellations around the room.
Optional
MLR8: Discussion Supports
To Gather
Geometry toolkits
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.
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
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.
Invite students to share their designs and also describe a transformation that takes the design to itself. Consider displaying their designs around the room.
Help us improve by sharing suggestions or reporting issues.
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.
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.
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.
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.