Rotate and Tessellate

10 min

Teacher Prep
Setup
Access to geometry toolkits. Distribute one half-sheet (that contains 7 shapes) to each student.

Narrative

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:

<p>Seven polygons.</p>

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 360÷6=60360 \div 6 = 60. For the other shapes, they can reason about angles that sum to 360 degrees, angles that sum to a line, and angles that sum to a known angle.

Launch

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.

Student Task

Your teacher will give you some shapes.

  1. 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?

  2. What are the measures of the angles in the

    1. Square?

    2. Hexagon?

    3. Parallelogram?

    4. Right triangle?

    5. Octagon?

    6. Pentagon?

Sample Response

  1. 6, 60∘60^\circ
  2. Measures of angles:
    1. Square: 90∘90^\circ
    2. Hexagon: 120∘120^\circ
    3. Parallelogram: 120∘120^\circ and 60∘60^\circ
    4. Right triangle: 45∘45^\circ and 90∘90^\circ
    5. Octagon: 135∘135^\circ
    6. Pentagon: 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ
Activity Synthesis (Teacher Notes)

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.

Anticipated Misconceptions

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.

Standards
Building On
  • 4.MD.C·Geometric measurement: understand concepts of angle and measure angles.
  • 4.MD.C·Geometric measurement: understand concepts of angle and measure angles.
  • 7.G.5·Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
  • 7.G.B.5·Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.