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Lesson 1Tiling the Plane

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

Learning Targets:

  • I can explain the meaning of area.

1.1 Which One Doesn’t Belong: Tilings

Which pattern doesn’t belong?

Four patterns of tiles labeled A, B, C, and D. Pattern A is all blue tiles, patter B is all yellow tiles, pattern C is a combination of blue and yellow tiles, and pattern D is a combination of blue tiles, yellow tiles, and blank spaces.

1.2 More Red, Green, or Blue?

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

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

Pattern A

Pattern B

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

open applet in presentation mode

Are you ready for more?

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.

Lesson 1 Summary

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 Terms

area

Area is the number of square units that covers 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.

A diagram showing what the area of a shape is
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.

Lesson 1 Practice Problems

  1. Which square—large, medium, or small—covers more of the plane? Explain your reasoning.

    A plane composed of a series of squares. There are 5 large squares, 10 medium squares, and 10 small squares.

  2. Draw three different quadrilaterals, each with an area of 12 square units.

    a blank grid

  3. Use copies of the rectangle to show how a rectangle could:

    a. tile the plane.

    a grid with blue rectangle that is 2 units tall and 3 units wide

    b. not tile the plane.

    a grid with blue rectangle that is 2 units tall and 3 units wide

  4. The area of this shape is 24 square units. Which of these statements is true about the area? Select all that apply.

    A figure with a top side of 4 units and a bottom side of 6 units. The lower portion of the figure has sides that are 2 units tall, and the upper portion has sides that 3 units tall. All angles are right angles.

    1. The area can be found by counting the number of squares that touch the edge of the shape.

    2. It takes 24 grid squares to cover the shape without gaps and overlaps.

    3. The area can be found by multiplying the side lengths that are 6 units and 4 units.

    4. The area can be found by counting the grid squares inside the shape.
    5. The area can be found by adding 4 \times 3 and 6 \times 2 .

  5. Here are two copies of the same figure. Show two different ways for finding the area of the shaded region. All angles are right angles.

    A figure with a bottom of 10 units, a right side of six units, and a left side that rises 3 units, then goes across 5 units, then goes up another 2 units, then across another 3 units, then up another 1 unit, and across another 2 units to connect to the right side. All angles are right angles.
    A figure with a bottom of 10 units, a right side of six units, and a left side that rises 3 units, then goes across 5 units, then goes up another 2 units, then across another 3 units, then up another 1 unit, and across another 2 units to connect to the right side. All angles are right angles.

  6. Which shape has a larger area: a rectangle that is 7 inches by \frac 34 inch, or a square with a side length of 2 \frac12 inches? Show your reasoning.