Tessellations

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5-9
Tessellations
Warm Up
Problem of the Day
Lesson Presentation
Course 3
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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
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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
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Learn to predict and verify patterns involving
tessellations.
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Insert Lesson Title Here
Vocabulary
tessellation regular tessellation semiregular
tessellation
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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.
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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.
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In a semiregular tessellation, two or more
regular polygons are repeated to fill the plane
and the vertices are all identical.
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Additional Example 1 Problem Solving Application
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.
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Additional Example 1 Continued
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
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Additional Example 1 Continued
There are two arrangements of three triangles and
two squares around a vertex.
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Additional Example 1 Continued
Repeat each arrangement around every vertex, if
possible, to create a tessellation.
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Additional Example 1 Continued
There are exactly two semiregular tessellations
that use triangles and squares.
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Additional Example 1 Continued
Every vertex in each arrangement is identical to
the other vertices in that arrangement, so these
are the only arrangements that produce
semiregular tessellations.
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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.
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Try This Example 2
Create a tessellation with quadrilateral IJKL.
There must be a copy of each angle of
quadrilateral IJKL at every vertex.
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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.
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Additional Example 3 Continued
Step 5 Use the figure to make a tessellation.
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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.
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Try This Example 3 Continued
Step 5 Use the figure to make a tessellation.
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Insert Lesson Title Here
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
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Insert Lesson Title Here
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