Lesson 1

Tiling the Plane

Let’s look at tiling patterns and think about area.

1.1: Which One Doesn’t Belong: Tilings

Which pattern doesn’t belong?

Four patterns of tiles labeled A, B, C, and D. 


1.2: More Red, Green, or Blue?

Your teacher will assign you to look at Pattern A or Pattern B.

In your pattern, which shapes cover more of the plane: blue rhombuses, red trapezoids, or green triangles? Explain how you know.

You may use the sliders and the shapes in this applet to help. Explore what you can see or hide, and what you can move or turn.

On graph paper, create a tiling pattern so that:

  • The pattern has at least two different shapes.
  • The same amount of the plane is covered by each type of shape.


In this lesson, we learned about tiling the plane, which means covering a two-dimensional region with copies of the same shape or shapes such that there are no gaps or overlaps. 

Then, we compared tiling patterns and the shapes in them. In thinking about which patterns and shapes cover more of the plane, we have started to reason about area.

We will continue this work, and to learn how to use mathematical tools strategically to help us do mathematics.

Glossary Entries

  • area

    Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

    For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.

    Figure A on the left composed of 8 identical shaded squares arranged in 3 rows. Figure B on the right consists of one square with a diagonal segment from corner to corner. Half of the square is shaded.
  • region

    A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.