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Lesson 2Tessellations of the Plane

Let’s explore geometric patterns!

2.1 Notice and Wonder: Polygon Patterns

What do you notice? What do you wonder?

Six squares with patterns. The first square shows green and blue L shaped polygons, stacked on top of each other to create a rectangle. The green and blue rectangles create diagonal lines. The second square has 10 rows of red and green alternating square tiles. The first row begins with a red tile and ends with a green tile. The second row begins with a green tile and ends with a red tile. The first and second row repeat for 10 rows. The third square has 10 rows of green and red arrows. The green arrows points to the right. The red arrows point to the left. Odd rows have green arrows. Even rows have red arrows. The fourth square has small green tiles with small red rhombuses at the corners of each square tile. There are 25 full green square tiles and 36 red rhombuses. The small red rhombuses are smaller than the green tiles, and do not touch each other. The fifth square has small green tiles with red parallelograms at the corners. The center of the parallelogram is aligned with the edge of the square; the left and right point of each parallelogram hits the center of the squares on either side of the edge. The vertex of each red parallelogram touches the vertex of the next red parallelogram. There are 25 full green square tiles and 36 red parallelograms. The sixth square has 16 full small square tiles and a border of 20 partial tiles. Each tile is comprised of 4 t shaped polygons, with the bottom point of each T facing into the center. The top left corner is a sideways yellow T. The top right corner is a blue T. The bottom right corner is a sideways red T. The bottom left corner is an upside down green T.

2.2 Tessellations

  1. With your partner, choose one of the six shapes in the toolbar that you will both use.

  • Select the shape tool by clicking on it. Create copies of your shape by clicking in the work space.

  • When you have enough to work with, click on the Move tool (the arrow) to drag or turn them.

  • If you have trouble aligning the shapes, right click to turn on the grid.

  • When you are finished making your tessellation, you can hide the points.

  1. Compare your tessellation to your partner’s. How are they similar? How are they different?

  2. If possible, make a third tessellation of the plane with your shape (different from the ones you and your partner already created). If not possible, explain why it is not possible.

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2.3 Describing a Tessellation

  1. Pick one of the figures and describe the tessellation. Your partner will identify which tessellation you are describing. Then trade roles so your partner describes the tessellation and you identify the figure.

    This should not be a alt text description. Tactile is needed.....Four tessellations.One tessellation has groups repeating rows of two rectangles. The first row has yellow and green alternating rectangles and the second row has blue and red alternating rectangles. Another tessellation has alternating blue and red alternating rectangles slanted upward and to the right. Below the red and blue rectangles are alternating yellow and green rectangles. A third tessellation has groups of three rectangles next to one another, sharing the long sides: the long sides are vertical. In addition to these groups of three rectangles, there are single rectangles with the long side lying horizontally. A fourth tessellation has alternating rectangles placed horizontally then vertically.

  2. You and your partner each have a card with a tessellation. Describe what is on your card so that your partner can produce the tessellation (this should be done so that you cannot see your partner’s work until it is complete).

  3. Check together to see if your partner’s tessellation agrees with your card and discuss any differences.

  4. Change roles so your partner describes a tessellation, which you attempt to produce.

  5. Check the accuracy of your construction and discuss any discrepancies.

2.4 Regular Tessellations

  1. For each shape (triangle, square, pentagon, hexagon, and octagon), decide if you can use that shape to make a regular tessellation of the plane. Explain your reasoning.

  2. For the polygons that do not work what goes wrong? Explain your reasoning.

2.5 Equilateral Triangle Tessellation

  1. What is the measure of each angle in an equilateral triangle? How do you know?
  2. How many triangles can you fit together at one vertex? Explain why there is no space between the triangles.
  3. Explain why you can continue the pattern of triangles to tessellate the plane.
  4. How can you use your triangular tessellation of the plane to show that regular hexagons can be used to give a regular tessellation of the plane?

2.6 Regular Tessellation for Other Polygons

  1. Can you make a regular tessellation of the plane using regular polygons with 7 sides? What about 9 sides? 10 sides? 11 sides? 12 sides? Explain.

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  1. How does the measure of each angle in a square compare to the measure of each angle in an equilateral triangle? How does the measure of each angle in a regular 8-sided polygon compare to the measure of each angle in a regular 7-sided polygon?

  2. What happens to the angles in a regular polygon as you add more sides?

  3. Which polygons can be used to make regular tessellations of the plane?

2.7 Triangle Tessellations

Your teacher will assign you one of the three triangles. Your goal is to find a tessellation of the plane with copies of the triangle.

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2.8 Quadrilateral Tessellations

  1. Can you make a tessellation of the plane with copies of the trapezoid? Explain.
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  1. Choose one of the other two quadrilaterals. Next, rotate the quadrilateral 180 degrees around the midpoint of each side. What do you notice?
  2. Can you make a tessellation of the plane with copies of the quadrilateral from the previous problem? Explain your reasoning.

2.9 Pentagonal Tessellations

  1. Can you tessellate the plane with copies of the pentagon? Explain. Note that the two sides making angle A are congruent.

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    Pause your work here.

  2. Take one pentagon and rotate it 120 degrees clockwise about the vertex at angle A, and trace the new pentagon. Next, rotate the pentagon 240 degrees clockwise about the vertex at angle A, and trace the new pentagon.

  3. Explain why the three pentagons make a full circle at the central vertex.

  4. Explain why the shape that the three pentagons make is a hexagon (i.e., the sides that look like they are straight really are straight).

Glossary Terms

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.

This diagram shows part of a tessellation.

a repeating pattern of octagons and squares