9-7
9-7
Tessellations
1. Plan
What You’ll Learn
Check Skills You’ll Need
• To identify transformations
Classify the polygon with the given number of sides.
in tessellations, and figures
that will tessellate
GO for Help
1. five pentagon
• To identify symmetries in
2. eight octagon
Lesson 3-5
. . . And Why
To identify a tessellation in
art, as in Example 1
1
dodecagon
3. twelve
2
Find the measure of an angle of each regular polygon.
tessellations
Objectives
4. triangle 60
5. quadrilateral 90
6. hexagon 120
7. octagon 135
8. decagon 144
9. 14-gon 154 27
Examples
1
New Vocabulary • tessellation • tiling • translational symmetry
• glide reflectional symmetry
2
3
1
To identify transformations in
tessellations, and figures that
will tessellate
To identify symmetries in
tessellations
Identifying the
Transformation in a
Tessellation
Determining Figures That
Will Tessellate
Identifying Symmetries in
Tessellations
Identifying Transformations in Tessellations
A tessellation, or tiling, is a repeating pattern of figures that completely covers a
plane, without gaps or overlaps. You can create tessellations with translations,
rotations, and reflections. You can find tessellations in art (see below), nature (cells
in a honeycomb), and everyday life (tiled floors).
Vocabulary Tip
A figure that creates a
tessellation is said to
tessellate.
1
EXAMPLE
Identifying the Transformation in a Tessellation
Art Identify a transformation and the repeating figures in this tessellation.
Math Background
The astronomer Johannes Kepler
is believed to have been the first
to investigate the possible ways of
covering a plane with regular
polygons. Analysis of tessellations
using more complicated figures
requires understanding of
isometry and symmetry.
More Math Background: p. 468D
Repeating figures
Lesson Planning and
Resources
SOURCE: © 1996 M. C. Escher Heirs/Cordon
Art, Baarn, Holland. All rights reserved.
See p. 468E for a list of the
resources that support this lesson.
The arrow shows a translation.
Quick Check
a–b. See left.
1 Identify a transformation and the repeating figures in each tessellation below.
a.
b.
PowerPoint
Bell Ringer Practice
1a. rotation; one fish
b. translation; horse
and rider
Check Skills You’ll Need
For intervention, direct students to:
Classifying Polygons
SOURCE: © 1996 M. C. Escher Heirs/Cordon
Art, Baarn, Holland. All rights reserved.
SOURCE: © 1996 M. C. Escher Heirs/Cordon
Art, Baarn, Holland. All rights reserved.
Lesson 3-5: Example 2
Extra Skills, Word Problems, Proof
Practice, Ch. 3
Polygon Angle-Sums
Lesson 9-7 Tessellations
Special Needs
Below Level
L1
Have students use pattern blocks to create tessellating
patterns. Encourage students to identify the figures
that tessellate.
learning style: tactile
515
Lesson 3-5: Examples 3, 5
Extra Skills, Word Problems, Proof
Practice, Ch. 3
L2
Have students search the classroom for tessellations.
Discuss their findings and whether they meet the
criteria for tessellations.
learning style: visual
515
2. Teach
Because the figures in a tessellation do not overlap or leave gaps, the sum of the
measures of the angles around any vertex must be 3608. If the angles around a
vertex are all congruent, then the measure of each angle must be a factor of 360.
Guided Instruction
2
EXAMPLE
Determining Figures That Will Tessellate
Careers
Determine whether a regular 18-gon tessellates a plane.
A tile setter uses tessellations to
design floors and other surfaces.
Although most tiles are square,
other shapes also are used. Have
students locate unusual tile or
mosaic patterns and copy them
to show the class.
a=
1
180(n 2 2)
n
180(18 2 2)
a=
18
a = 160
Quick Check
EXAMPLE
EXAMPLE
Key Concepts
Visual Learners
Additional Examples
2 Determine whether a regular
15-gon tessellates a plane. Explain.
No; 156 is not a factor of 360.
2 Explain why you can tessellate a plane with an equilateral triangle.
The interior ' of an equilateral k measure 60.
60 divides 360, so it will tessellate.
A figure does not have to be a regular polygon to tessellate.
Every triangle tessellates.
Theorem 9-7
Every quadrilateral tessellates.
2
1
Identifying Symmetries in Tessellations
1 Identify the repeating figures
and a transformation in the
tessellation.
regular octagon and square;
translation
The tessellation with regular hexagons at the right has
reflectional symmetry in each of the blue lines. It has
rotational symmetry centered at each of the red points.
The tessellation also has translational symmetry and
glide reflectional symmetry, as shown below.
Real-World
Connection
Careers Physicists apply the
symmetries of tessellations to
study subatomic particles.
Here they use a particle
detector to study quarks.
516
Translational Symmetry
Glide Reflectional Symmetry
A translation maps the
tessellation onto itself.
A glide reflection maps the
tessellation onto itself.
Chapter 9 Transformations
Advanced Learners
English Language Learners ELL
L4
After students complete Example 2, have them
determine which regular n-gons tessellate a plane,
and prove their findings.
516
Simplify.
Theorem 9-6
To help students see why the
angle measure of a regular
polygon must be a factor of 360
for the polygon to tessellate a
plane, have them use geometry
drawing software or a protractor
to construct adjacent 35° angles
with the same vertex. They will
find that ten 35° angles leave a
gap of 10°, and eleven adjacent
35° angles overlap.
PowerPoint
Substitute 18 for n.
Since 160 is not a factor of 360, the 18-gon will not tessellate.
Ask students to name the simplest
transformation, not a composition
of two reflections.
2
Use the formula for the measure of an angle of a regular polygon.
learning style: verbal
In Example 3, students may have difficulty describing
symmetries. Pair students with partners who can help
them express the relationships they observe.
learning style: verbal
3
Guided Instruction
EXAMPLE
Identifying Symmetries in Tessellations
3
List the symmetries in the tessellation.
Rotational symmetry centered at each red point
Translational symmetry (blue arrow)
EXAMPLE
Remind students that a drawing
can only suggest that tessellations,
like planes, extend without bound
in all directions.
Hands-On Activity
Quick Check
After students have designed
their figures, have them trace the
outlines on lightweight cardboard,
cut out the figures, and trace
around the cutouts to create their
tessellations.
3 List the symmetries in the
tessellation at the right.
line symmetry, rotational
symmetry, glide reflectional
symmetry, translational
symmetry
PowerPoint
Additional Examples
3 List the symmetries in the
tessellation.
The following Investigation shows the steps for making creative tessellations.
Hands-On Activity: Creating Tessellations
• Draw a 1.5-inch square on a blank piece of
paper and cut it out.
• Draw a curve joining two consecutive vertices.
• Cut along the curve you drew
and slide the cutout piece to
the opposite side of the square.
Tape it in place.
point symmetry centered at
any vertex, translational
symmetry
Cut
Slide
Tape
Check students’ work.
• Repeat this process using the other two
opposite sides of the square.
Tape
• Rotate the resulting figure. What does
your imagination suggest it looks like?
Is it a penguin wearing a hat or a knight on
horseback? Could it be a dog with floppy
ears? Draw the image on your figure.
Slide
Closure
Use angle measures to explain
how an arrangement of regular
12-gons and equilateral triangles
tessellates a plane. An angle of
an equilateral triangle measures
60, and an angle of a regular
12-gon measures 150. Because
60 ± 150 ± 150 ≠ 360, two
12-gons share a vertex with
each vertex of the triangles.
Cut
• Create a tessellation
using your figure.
Lesson 9-7 Tessellations
Resources
• Daily Notetaking Guide 9-7 L3
• Daily Notetaking Guide 9-7—
L1
Adapted Instruction
517
517
EXERCISES
3. Practice
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
Assignment Guide
A
1 A B 1-10, 15-22
Practice by Example
Example 1
2 A B
11-13, 23-26
C Challenge
27-41
Test Prep
Mixed Review
42-46
47-56
Homework Quick Check
GO for
Help
(page 515)
2.
3.
4.
1. yes; translation;
two # rectangles
2. yes; translation;
two ? and a
rhombus with a
flower in it
Error Prevention!
3. yes; translation;
four rectangles in a
square shape
(see overlay)
4. no
Example 2
(page 516)
Exercise 14 The text definition of
Determine whether each figure will tessellate a plane.
5. equilateral triangle yes 6. square yes
8. regular heptagon no
tessellation is that of a periodic
tessellation. The idea of
tessellation is more general. This
exercise shows a tessellation that
is not periodic, but still involves a
figure that repeats.
GPS Guided Problem Solving
1.
1–4. Answers may vary.
Samples are given.
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 12, 16, 22, 26.
Exercises 11–14 Encourage
students to trace the tessellations,
place dots at the center of figures
or “spaces,” and then test for
rotational symmetry by turning
their tracings. Encourage them to
draw lines through vertices and
midpoints of figures to test for
reflectional symmetry.
Does the picture show a tessellation of repeating figures? If so, identify a
transformation and the repeating figure.
Example 3
(page 517)
7. regular pentagon no
9. regular octagon no
List the symmetries in each tessellation. 11–14. See margin.
11.
12.
13.
14.
L3
L4
Enrichment
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
B
L3
Date
Practice 9-5
Apply Your Skills
Use each figure to create a tessellation on dot paper. 15–17. See margin.
Trigonometry and Area
15.
Find the area of each polygon. Round your answers to the nearest tenth.
1. an equilateral triangle with apothem 5.8 cm
16.
17.
2. a square with radius 17 ft
3. a regular hexagon with apothem 19 mm
4. a regular pentagon with radius 9 m
5. a regular octagon with radius 20 in.
6. a regular hexagon with apothem 11 cm
7. a regular decagon with apothem 10 in.
8. a square with radius 9 cm
Find the area of each triangle. Round your answers to the nearest tenth.
9.
6.5 m
63
13 m
10.
11.
9 mi
10 km
38
18 km
42
10 mi
12.
13.
© Pearson Education, Inc. All rights reserved.
518
14.
28 in. 59
6 mm
34 in.
Chapter 9 Transformations
32 in.
46
4.5 mm
54
26 in.
15. 10 cm
16.
35
19 cm
65
5 ft
4 ft
17.
15 m
46
15 m
Find the area of each regular polygon to the nearest tenth.
18. a triangular dog pen with apothem 4 m
11. rotational, reflectional,
glide reflectional, and
translational
13. rotational, reflectional,
glide reflectional, and
translational
12. rotational, point,
reflectional, glide
reflectional, and
translational
14. rotational and
reflectional
19. a hexagonal swimming pool cover with radius 5 ft
20. an octagonal floor of a gazebo with apothem 6 ft
21. a square deck with radius 2 m
22. a hexagonal patio with apothem 4 ft
518
10. regular nonagon no
15.
GO
nline
Homework Help
18. Multiple Choice Which jigsaw puzzle piece can tessellate a plane using only
translation images of itself? C
Visit: PHSchool.com
Web Code: aue-0906
22. A regular polygon
with more than 6
sides must have l
measures greater
than 120, and at least
3 polygons must meet
at each vertex. The
sum of 3 or more '
with measures greater
than 120 S 360. So
the 3 regular polygons
are 3-, 4-, and 6-sided,
since their int. l
measures divide 360.
Diversity
Ukrainian Easter eggs, or pysanky,
have tessellations painted on their
surfaces. The world’s largest
pysanka is in Vegreville, Canada.
Exercise 18 Students can trace
Show how to tessellate with each figure described below. Try to draw two different
tessellations. If you think that two are not possible, explain.
19. a scalene triangle
20. the pentagon at the right
19–21. See
back of book.
each puzzle piece and test
whether it tessellates using only
translations.
Exercise 22 Point out that these
21. a quadrilateral with no sides parallel or congruent
22. Writing A pure tessellation is a tessellation made up of congruent copies of one
figure. Explain why there are three, and only three, pure tessellations that use
regular polygons. (Hint: See Exercises 5–10.) See left.
three examples are called regular
tessellations. Compare this
exercise with Exercises 23 and 24,
which investigate semiregular
tessellations.
Exercise 24 Challenge students
Decide whether a semiregular tessellation (see photo) is possible using the given
pair of regular polygons. If so, draw a sketch.
23.
24.
no
See margin.
to find more semiregular
tessellations.
Alternate Method
Exercise 40 Have students use
geometry software to perform
the translation and rotations.
Challenge them to use their
observations to explain why
every quadrilateral tessellates.
Can each set of polygons be used to create a tessellation? If so, draw a sketch.
25.
25–26. See
margin.
26.
60°
GPS
A semiregular tessellation is
made from two or more
regular polygons.
25. yes;
26. yes;
List the symmetries of each tessellation. 27–28. See margin.
27.
C
Challenge
28.
Copy the Venn diagram. Write each exercise
number in the correct region of the diagram.
29. scalene triangle
30. obtuse triangle
31. equilateral n
32. isosceles n
33. kite
34. rhombus
35. square
29–39.
See margin.
Figures that tessellate
36. regular pentagon
37. regular hexagon
38. regular octagon
lesson quiz, PHSchool.com, Web Code: aua-0907
16.
Regular
figures
17.
Polygons
27. reflectional, glide
reflectional, and
translational
28. rotational, point,
reflectional, glide
reflectional, and
translational
29– 39.
Regular
Figures
29 33
27
35 28 30
31 32
39. regular decagon
Lesson 9-7 Tessellations
24. yes;
Polygons
34
36 37
519
Figures That Tessellate
40. a – c. Drawings may vary.
Sample:
C
D
A
M
(a)
N
B
(b)
(c)
519
4. Assess & Reteach
PowerPoint
Lesson Quiz
Determine whether each figure
will tessellate a plane.
40d. Yes, ABCD
tessellates; the
sum of the
measures of the '
of a quad. is 360.
Copies of the quad.
can be arranged so
that the four '
share a vertex. The
quad. fills the plane.
40. On graph paper, draw quadrilateral ABCD with no two sides congruent.
Locate M, the midpoint of AB, and N, the midpoint of BC.
a–c.
a. Draw the image of ABCD under a 1808 rotation about M.
See margin.
b. Draw the image of ABCD under a 1808 rotation about N.
c. Draw the image of ABCD under the translation that maps D to B.
d. Make a conjecture about whether your quadrilateral tessellates, using the
pattern in parts (a)–(c). Justify your answer. See left.
41. List steps (like those in Exercise 40) that suggest a way to tessellate with any
scalene triangle. Then list a second set of steps that suggest another way.
See margin.
1. regular octagon no
2. regular hexagon yes
3.
Test Prep
Multiple Choice
42. Which figure will NOT tessellate a plane? B
A.
B.
C.
D.
yes
Use the tessellation below for
Exercises 4 and 5.
43. You can tessellate a plane using a regular octagon together with which
other type of regular polygon? G
F. triangle
G. square
H. pentagon
J. hexagon
44. Which is NOT a symmetry for the tessellation? C
A. line symmetry
B. translational symmetry
C. rotational symmetry
D. glide reflectional symmetry
Short Response
4. List the repeating figures.
regular pentagon, isosceles
triangle,
concave quadrilateral
45. Is it possible to tile a plane with regular pentagons? Justify your answer.
See left below.
46. Unit squares form this tessellation. Tell whether
this tessellation has each type of symmetry
(line, point, rotational, translational, or glide
reflectional). Explain. See margin pp. 520–521.
Extended Response
5. List the symmetries. vertical
line symmetry, translational
symmetry
Mixed Review
Alternative Assessment
Have each student make 12
construction-paper copies of a
quadrilateral whose sides differ in
length, arrange the quadrilaterals
to form a tessellation, and then
explain in writing what a
tessellation is and what theorem
guarantees that the figure will
tessellate the plane.
Lesson 9-6
GO for
Help
Lesson 8-1
520
Algebra Find the value of x.
2
x 48.
!89 5
x
49.
10!2
x
14
x
6
The lengths of two sides of a triangle are given. What are the possible lengths for
the third side?
51. 16 in., 26 in.
10 R s R 42
54. 2 cm, 7 cm
5RsR9
520
x
50.
4!10
20
8
Lesson 5-5
41. Answers may vary. Sample:
Draw kABC. Locate M,
the mdpt. of AB, and N,
the mdpt. of BC. Draw the
images of kABC under
180 rotations about M
and N. Draw the image
of kABC under the
translation that maps A
to C. 2nd way: Draw kABC.
Draw the reflection image
47. A triangle has vertices A(3, 2), B(4, 1), and C(4, 3). Find the coordinates of
the images of A, B, and C for a glide reflection with translation (x, y) S
(x, y + 1) and reflection line x = 0. A9(–3, 3), B9(–4, 2), C9(–4, 4)
52. 19.5 ft, 20.5 ft
1 R s R 40
55. 412 yd, 8 yd
1
31
2 R s R 122
53. 9 m, 9 m
0 R s R 18
56. 1 km, 2 km
1RsR3
Chapter 9 Transformations
of pt. C over AB, C. Now
use the steps from Ex.
38 for quad. ACBC.
46. [4] line symmetry: any
vert. or horiz. line, a
diag. line drawn through a
square’s corners, or a line
drawn through the # bis. of a
square’s side; rotational
symmetry: 90 and point
about any square’s center;
translational: vertically or
horizontally 1 unit; glide refl.:
translate 1 unit and reflect
through a line of symmetry.
(OR equivalent explanation)
Test Prep
Checkpoint Quiz 2
Lessons 9-4 through 9-7
Resources
The two figures in each pair are congruent. Is one figure a translation image, a
rotation image, or a reflection image of the other? Explain.
P
P
2.
Rotation; the
image appears
rotated about 90.
So
1.
So
Reflection; the
orientation is reversed.
The blue figure is a dilation image of the red figure. Describe the dilation.
5. line, point
3.
y
4.
5
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p.5
• Test-Taking Strategies, p. 5
• Test-Taking Strategies with
Transparencies
y
4
x
x
-2
O 2
O 2
-2
enlargement; center (0, 0); scale factor 2 reduction; center (0, 0); scale factor 0.5
Tell what type(s) of symmetry (line, rotational, or point) each figure has. For line
symmetry, sketch the figure and the line(s) of symmetry. For rotational symmetry,
state the angle of rotation.
7. line, rotational: 120
5.
See
left.
6.
7.
Use this Checkpoint Quiz to check
students’ understanding of the
skills and concepts of Lessons 9-4
through 9-7.
Resources
Grab & Go
• Checkpoint Quiz 2
8.
See left.
point
point
List the symmetries of each tessellation. 9–10. See left.
9.
9. rotational, reflectional,
glide reflectional, and
translational
10.
10. rotational, point,
reflectional, glide
reflectional, and
translational
A P int in Time
1500
1600
1700
1800
1900
2000
A
mosaic is a picture or design made by setting tiny pieces of glass, stone, or
other materials in clay or plaster. A mosaic may be a tessellation. Most mosaics,
however, do not have a repeating pattern of figures. Mosaics go back at least
6000 years to the Sumerians, who used tiles to both decorate and reinforce walls.
During 100 and 200 A.D., Roman architects used two million tiles to create the
magnificent mosaic of Dionysus in Germany. In the years 1941–1951 Mexican
artist Juan O’Gorman covered all four sides of a 10-story library in Mexico with
7.5 million stones—the largest mosaic ever. It depicts Mexico’s cultural history.
PHSchool.com
For: Information about mosaics
Web Code: afe-2032
Lesson 9-7 Tessellations
[3] 2 or 3 correct answers
and explanations OR
3 or 4 correct answers
with 2 or 3 incorrect
explanations
[2] 1 or 2 correct answers
and explanations OR
2 or 3 correct answers
with 1 or 2 incorrect
explanations
521
[3] incorrect explanations
and answers
521