pre-algebra 5-9 tessellations 5-9 tessellations pre-algebra warm up warm up problem of the day...
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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°