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Mathematics LibreTexts

1.1.1: Tiling the Plane

  • Page ID
    39427
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    Lesson

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

    Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Tilings

    Which pattern doesn’t belong?

    clipboard_ebeae8ab4fac033446cce314be4d5194e.png
    Figure \(\PageIndex{1}\): Four patterns of tiles labeled A, B, C, and D. Pattern A is all blue tiles, pattern 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.

    Exercise \(\PageIndex{2}\): 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.

    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.

    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 Entries

    Definition: 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 \(\frac{1}{2}\) square unit.

    clipboard_edee6cbe6defa851c039ab6d162f09df2.png
    Figure \(\PageIndex{2}\)

    Definition: 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.

    Practice

    Exercise \(\PageIndex{3}\)

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

    clipboard_ef755f1db69cea76f9a8c2399e26718fd.png
    Figure \(\PageIndex{3}\)

    Exercise \(\PageIndex{4}\)

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

    clipboard_ecda61aad4d7af43f67404d7e446f437c.png
    Figure \(\PageIndex{4}\)

    Exercise \(\PageIndex{5}\)

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

    1. tile the plane.
    clipboard_e8e6d2d64236978dd8cc874f483d2171d.png
    Figure \(\PageIndex{5}\)
    1. not tile the plane.
    clipboard_e6dde2fde4ddbedad464a3a0fb49ce619.png
    Figure \(\PageIndex{6}\)

    Exercise \(\PageIndex{6}\)

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

    clipboard_e11d8658265673f5cea4cd2035d0e4fd2.png
    Figure \(\PageIndex{7}\): A figure on a grid. 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 sides 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 \(6\times 2\).

    Exercise \(\PageIndex{7}\)

    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.

    clipboard_e7bf46c82cce5c7078bfe1ad031645fd8.png
    Figure \(\PageIndex{8}\): 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.
    clipboard_ea762b8870df90e861d2fd75b78f20796.png
    Figure \(\PageIndex{9}\): 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.

    Exercise \(\PageIndex{8}\)

    Which shape has a larger area: a rectangle that is 7 inches by \(\frac{3}{4}\) inch, or a square with side length of \(2\frac{1}{2}\) inches? Show your reasoning.