Pre-Algebra

5-9 Tessellations

5-9 Tessellations

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

5-9 Tessellations

Warm UpIdentify each polygon.

1. polygon with 10 sides

2. polygon with 3 congruent sides

3. polygon with 4 congruent sidesand no right angles

decagon

equilateral triangle

Pre-Algebra

5-9 Tessellations

rhombus

Pre-Algebra

5-9 Tessellations

Problem of the DayIf each of the capital letters of the alphabet is rotated 180° around its center, which of them will look the same? H, I, N, O, S, X, Z

Pre-Algebra

5-9 Tessellations

Learn to predict and verify patterns involving tessellations.

Pre-Algebra

5-9 Tessellations

Vocabulary

tessellation

regular tessellation

semiregular tessellation

Pre-Algebra

5-9 Tessellations

Fascinating designs can be made by repeating a figure or group of figures. These designs are often used in art and architecture.

A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps is a tessellation.

Pre-Algebra

5-9 Tessellations

In a regular tessellation, a regular polygon is repeated to fill a plane. The angles at each vertex add to 360°, so exactly three regular tessellations exist.

Pre-Algebra

5-9 Tessellations

In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical.

Pre-Algebra

5-9 Tessellations

11 Understand the Problem

Find all the possible semiregular tessellations that use triangles and squares.

List the important information:

• The angles at each vertex add to 360°.

• All the angles in a square measure 90°.

• All the angles in an equilateral triangle measure 60°.

Additional Example 1: Problem Solving Application

Pre-Algebra

5-9 Tessellations

22 Make a Plan

Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation.

Additional Example 1 Continued

6 triangles, 0 squares 6(60°) = 360° regular

3 triangles, 2 squares 3(60°) + 2(90°) = 360°

0 triangles, 4 squares 4(90°) = 360° regular

Pre-Algebra

5-9 Tessellations

Solve33

There are two arrangements of three triangles and two squares around a vertex.

Additional Example 1 Continued

Pre-Algebra

5-9 Tessellations

Repeat each arrangement around every vertex, if possible, to create a tessellation.

Additional Example 1 Continued

Solve33

Pre-Algebra

5-9 Tessellations

There are exactly two semiregular tessellations that use triangles and squares.

Solve33

Additional Example 1 Continued

Pre-Algebra

5-9 Tessellations

Look Back44

Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Additional Example 1 Continued

Pre-Algebra

5-9 Tessellations

Additional Example 2: Creating a Tessellation

Create a tessellation with quadrilateral EFGH.

There must be a copy of each angle of quadrilateral EFGH at every vertex.

Pre-Algebra

5-9 Tessellations

Try This: Example 2

Create a tessellation with quadrilateral IJKL.

There must be a copy of each angle of quadrilateral IJKL at every vertex.

I

J

K

L

Pre-Algebra

5-9 TessellationsAdditional Example 3: Creating a Tessellation by

Transforming a polygon

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Pre-Algebra

5-9 Tessellations

Additional Example 3 Continued

Step 5: Use the figure to make a tessellation.

Pre-Algebra

5-9 Tessellations

Try This: Example 3Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Pre-Algebra

5-9 Tessellations

Step 5: Use the figure to make a tessellation.

Try This: Example 3 Continued

Pre-Algebra

5-9 Tessellations

Lesson Quiz: Part 1

1. Find all possible semiregular tessellations that use squares and regular hexagons.

2. Explain why a regular tessellation with regular octagons is impossible. Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

none

Pre-Algebra

5-9 Tessellations

Lesson Quiz: Part 2

3. Can a semiregular tessellation be formed using a regular 12-sided polygon and a regular hexagon? Explain.

No; a regular 12-sided polygon has angles that measure 150° and a regular hexagon has angles that measure 120°. No combinations of 120° and 150° add to 360°