1. 5-9 Tessellations Warm Up Problem of the Day Lesson Presentation
Pre-Algebra

2. 5-9 Tessellations Warm Up Identify each polygon.
Pre-Algebra
5-9
Tessellations
Warm Up
Identify each polygon.
1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sides
and no right angles
decagon
equilateral triangle
rhombus

3. Problem of the Day
If 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

4. Learn to predict and verify patterns involving tessellations.

5. Vocabulary
tessellation
regular tessellation
semiregular tessellation

6. 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.

7. 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.

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

9. Understand the Problem
Additional Example 1: Problem Solving Application
Find all the possible semiregular tessellations that use triangles and squares. 1
Understand the Problem
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°.

10. Additional Example 1 Continued2
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.
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

11. Additional Example 1 Continued
Solve 3
There are two arrangements of three triangles and two squares around a vertex.

12. Additional Example 1 Continued
Solve 3
Repeat each arrangement around every vertex, if possible, to create a tessellation.

13. Additional Example 1 Continued
Solve 3
There are exactly two semiregular tessellations that use triangles and squares.

14. Additional Example 1 Continued
Look Back 4
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

15. 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.

16. Try This: Example 2
Create a tessellation with quadrilateral IJKL. J K L I
There must be a copy of each angle of quadrilateral IJKL at every vertex.

17. Additional 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.

18. Additional Example 3 Continued
Step 5: Use the figure to make a tessellation.

19. Try This: Example 3
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.

20. Try This: Example 3 Continued
Step 5: Use the figure to make a tessellation.

21. 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.
none
Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

22. 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°