Escher-Type Translation
Tessellations
Objective To create nonpolygonal, Escher-type
ttranslation tessellations.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Recognize patterns involving angle
measures around any vertex point
in tessellations. [Geometry Goal 1]
• Apply isometry transformations to
create nonpolygonal, Escher-type
translation tessellations. [Geometry Goal 3]
Key Activities
Students create nonpolygonal, Escher-type
translation tessellations.
Ongoing Assessment:
Recognizing Student Achievement
Use Mental Math and Reflexes. [Number and Numeration Goal 6]
Key Vocabulary
translation tessellation
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
Identifying Isometry
Transformations
Math Journal 2, p. 374
Student Reference Book, pp. 180
and 181 (optional)
Students practice identifying figures
resulting from translations, reflections,
and rotations. They use absolute value
to find distances between points on
the coordinate grid.
Math Boxes 10 2
Math Journal 2, p. 375
Students practice and maintain skills
through Math Box problems.
Study Link 10 2
Math Masters, p. 334
Students practice and maintain skills
through Study Link activities.
Materials
Math Journal 2, p. 373
Student Reference Book, pp. 359 and 360
Study Link 10 1
squares of cardstock (3" by 3") scissors tape white construction paper (11" by 17")
markers, crayons, or colored pencils
Differentiation Options
READINESS
Exploring Quadrangle Tessellations
Math Masters, p. 335
Student Reference Book, p. 165 (optional)
cardstock protractor scissors
Students explore whether convex
quadrangles tessellate.
ENRICHMENT
Creating Escher-Type Tessellations
squares of cardstock (3" by 3") scissors tape
Students create a template for a
rotation tessellation.
ENRICHMENT
Creating Quilt Patterns
Eight Hands Round: A Patchwork Alphabet
white construction paper (11" by 17") markers, crayons, or colored pencils
Students read about the tessellations
involved in quilt designs.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, pp. 65, 148, 149, 227,
and 230
Students identify and perform isometry
transformations.
Advance Preparation
For Part 1, cut 3" by 3" squares from index cards or the blank cardstock pages at the back of students’
journals. Review page 360 in the Student Reference Book before teaching this lesson.
For the optional Enrichment activity in Part 3, obtain the book Eight Hands Round: A Patchwork
Alphabet by Ann Whitford Paul (HarperCollins, 1991).
Teacher’s Reference Manual, Grades 4–6 pp. 196–199, 201–206
886
Unit 10
Geometry Topics
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8
Content Standards
Getting Started
6.NS.6b, 6.NS.8, 6.G.3
Mental Math and Reflexes Math Message
Read page 359 in your Student Reference Book.
Identify the transformation(s) that move one sea horse
figure onto another.
Students identify the number that is closer to 3.
2_
or 2_
2_
8
12 12
59
70 59
_
or _ _
47
42 47
_
or _ _
11
14 14
_
or _ _
3_
or 2_
2_
7
3 3
34
9 34
_
or 2_ _
7
16
3
11
11
15 16
2 2
20
3
11
20 20
4
4
Study Link 10 1 Follow-Up
11 11
Briefly go over the answers. Ask students to share
their tessellations from Problem 3.
Ongoing Assessment:
Recognizing Student Achievement
Mental Math
and
Reflexes
Use Mental Math and Reflexes to assess students’ ability to compare
rational numbers. Students are making adequate progress if they are able
to identify the number in each pair that is closer to 3.
[Number and Numeration Goal 6]
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 359)
Discuss any questions or comments that students may have
about the information on page 359 of the Student Reference Book.
Write the key ideas from the discussion on the board. Have
students share the isometry transformations they identified that
move one sea horse figure onto another.
NOTE The translation technique described
on Student Reference Book, page 360 can
also be used with parallelograms and regular
hexagons. However, because a regular
hexagon has three pairs of parallel sides,
three sets of opposite curves must be drawn.
Student Page
Date
Time
LESSON
10 2
▶ Creating an Escher-Type
Translation Tessellation
INDEPENDENT
ACTIVITY
PROBLEM
P
PR
PRO
ROBL
ROBL
BLE
B
LE
L
EM
SOL
SOL
SO
SOLVING
OLV
VIN
NG
My Tessellation
On a separate sheet of paper, create an Escher-type translation tessellation using
the procedure described on page 360 of the Student Reference Book. Experiment
with several tessellations until you create one that you especially like.
Trace your final tessellation template in the space below.
Answers vary.
(Math Journal 2, p. 373; Student Reference Book, p. 360)
Art Link Students follow the procedure on page 360 of the
Student Reference Book to create an Escher-type translation
tessellation. A translation tessellation is created by translating
curves from one side of a shape to the opposite side. In this case,
the shape is a square. Students will trace the template and then
translate it to create the next interlocking piece in the tessellation.
In the completed drawing, all figures will face the same direction.
In the space below, use your tessellation template to record what your tessellation
looks like. Add details or color to your final design.
Answers vary.
Encourage students not to settle for the first tessellation template
they create, but to spend some time experimenting until a
recognizable figure appears. Have students record their
tessellation templates and tessellations in their journals.
Math Journal 2, p. 373
370_387_EMCS_S_G6_U10_576442.indd 373
2/22/11 4:39 PM
Lesson 10 2
887
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Translations, Reflections, Rotations
10 2
Use the grid to help you answer the questions below.
180 181
y
▶ Identifying Isometry
8
7
1
6
E
5
2
4
D
Transformations
3
2
1
A
8
7
6
5
4
3
INDEPENDENT
ACTIVITY
2
1 0
1
4
2
1
2
3
4
5
6
7
(Math Journal 2, p. 374; Student Reference Book, pp. 180 and 181)
x
8
3
3
B
1.
Students identify figures on a coordinate grid that are the result
of a translation, a reflection, or a rotation. They find distances
between vertices of the figures. Before they begin, remind students
how to use absolute value to find distances between points on a
coordinate grid.
C
Identify which triangle results from performing the following transformations on the
shaded right triangle.
2
4
3
1
a.
b.
c.
d.
2.
4
A 90° counterclockwise rotation around point (-5,1)
A reflection over the x-axis
A reflection over the y-axis, followed by a reflection over the x-axis
A reflection over the y-axis, followed by a translation 4 units up
Use absolute value to find the distance between the following points. Write a
number sentence to show how you found your answer.
5; |1 - (-4)| = |5| = 5
2; |-1 - 1| = |-2| = 2
c. C and D 9; |-4 - 5| = |-9| = 9
d. D and E 6; |1 - (-5)| = |6| = 6
a.
A and B
b.
B and C
▶ Math Boxes 10 2
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 375)
Math Journal 2, p. 374
370_387_EMCS_S_G6_U10_576442.indd 374
2/21/11 4:41 PM
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 10-4.
Writing/Reasoning Have students write a response to
the following: Explain how you estimated the product in
Problem 2b. Sample answer: First, I rounded the factors
to 4 and 67. Then I applied the distributive property and
multiplied 4 ∗ (60 + 7) to get 240 + 28, or 268.
▶ Study Link 10 2
(Math Masters, p. 334)
Home Connection Students plot points on a
coordinate grid to draw the images of figures resulting
from translations.
Student Page
Date
Time
LESSON
10 2
1.
Math Boxes
Apply the order of operations to evaluate
each expression.
a.
15 - 3.3 ∗ 4 =
20
_
b. 4
2.
1.8
9
∗ 5 + (-8) ∗ 2 =
0.12
-28
d. 8 ∗ (2 + -5) - 4 =
_=
58
e. 7 ∗ 3 c.
10
2
5.25 ÷ 2.003
b.
4.29 ∗ 67.1
c.
80.25 ÷ 18.93
d.
52.31 ∗ 19.9
2.5
280
4
About
About 1,040
About
37–45
4.
7,400,000,000,000
d.
1,234.5 million
READINESS
Write the following in standard notation.
▶ Exploring Quadrangle
53,800,000
0.00691
3.04
∗
=
c. 3.04 10
990,110 = 9.9011 ∗ 10
d.
= 0.0000072
e. 7.2 ∗ 10
2.5 million
300
b. 0.3 thousand
2,500,000
c.
3 Differentiation Options
About
247
Convert between number-and-word and
standard notations.
a.
a.
0.01 + 0.01 ∗ 10 + 0.01 =
2
3.
Estimate each quotient or product.
Sample estimates:
a.
5.38 ∗ 107 =
b.
6.91 ∗ 10-3 =
5
(Math Masters, p. 335; Student Reference Book, p. 165)
-6
1,234,500,000
5.
7
Write an algebraic expression for the
following situation.
6.
Rafael is twice as old as his brother Jorge
was 3 years ago. Jorge is j years old now.
How old is Rafael?
Expression
2( j - 3) = r
8
Fill in the blanks to complete each
number sentence.
8
5)
6 ) - (3 ∗ 6 )
c. (9 ∗ 50) + (9 ∗ 4 ) = 9(50 + 4)
d. (7 ∗ 20) - (7 ∗ 3) = 7 (20 - 3 )
a.
8(40 + 5) = (
b.
(10 - 3)6 = (10 ∗
240
∗ 40) + (8 ∗
248 249
Math Journal 2, p. 375
370_387_EMCS_S_G6_U10_576442.indd 375
888
Unit 10
5–15 Min
Tessellations
0
7.4 trillion
PARTNER
ACTIVITY
2/22/11 4:39 PM
Geometry Topics
Students tessellate convex quadrangles they made to try to
answer the question: Do all convex quadrangles tessellate?
When they have completed this exploration, discuss the results.
Consider extending the investigation by asking the question:
Do all concave quadrangles tessellate? Yes
Study Link Master
Because the sum of the measures of the four angles of a convex
quadrangle is 360°, any convex quadrangle can be tessellated.
Emphasize to students that when tessellating a quadrangle, they
should make sure all four angles meet at each vertex point and
the sum of the angle measures about each vertex point is 360°.
If necessary, have students refer to Student Reference Book,
page 165, to review convex and concave polygons.
Name
Date
STUDY LINK
Time
Translations
10 2
180 181
Plot and label the vertices of the image that would result from each translation.
One vertex of each image has already been plotted and labeled.
y
y
1.
10
9
8
7
6
5
4
D
A
horizontal
translation
B
3
2
1
C
0
10
9
8
7
6
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10
x
A'
D'
B'
C'
image
y
PARTNER
ACTIVITY
ENRICHMENT
▶ Creating Escher-Type
x
0 0 1 2 3 4 5 6 7 8 9 10
preimage
y
2.
10
9
8
7
6
5
4
3
2
1
5–15 Min
Tessellations
0
I
10
9
8
7
6
5
4
3
2
1
J
H
vertical
translation
E
G
F
0 1 2 3 4 5 6 7 8 9 10
x
0
I'
J'
H'
E'
F
F'
G'
0 1 2 3 4 5 6 7 8 9 10
preimage
x
image
y
y
3.
10
9
8
7
6
5
4
3
2
1
Art Link To extend students’ knowledge of Escher-type
tessellations, help them create a tessellating shape. Using
the example below as a reference, demonstrate how to modify
opposite sides of a square and rotate each modification about a
vertex to an adjacent side. Have students work with a partner to
practice the modifying-rotating technique on the opposite sides of
a square or an equilateral triangle. When students are satisfied
with their shape, have them check whether it tessellates.
0
L
N
M
O
K
0 1 2 3 4 5 6 7 8 9 10
L' M ' N'
L'
10
9
8
7
6
5
4
3
2
1
diagonal
translation
x
0
O'
O'
K'
K'
0 1 2 3 4 5 6 7 8 9 10
preimage
x
image
Practice
4.
25.6
_
32
0.8
5.
102.4
_
64
1.6
6.
41.83
_
4.7
8.9
7.
67.32
_
13.2
5.1
Math Masters, p. 334
329-349_EMCS_B_G6_MM_U10_576981.indd 334
2/26/11 3:18 PM
Example:
A
1
D
2
4
B
3
C
1 A 2
3 C 4
Teaching Master
Name
LESSON
10 2
▶ Creating Quilt Patterns
5–15 Min
Literature Link To further explore the uses of tessellations,
have students read Eight Hands Round: A Patchwork
Alphabet by Ann Whitford Paul (HarperCollins, 1991). Then
assign students to partnerships and provide each with 11" by 17"
white construction paper and markers, crayons, or colored pencils.
Ask students to recreate one of the quilts from the book.
▶ 5-Minute Math
5–15 Min
To offer more practice identifying and performing isometry
transformations, see pages 65, 148, 149, 227, and 230 of
5-Minute Math.
Answers vary.
1.
Draw a convex quadrangle on a piece of cardstock paper.
2.
Measure the angles of your quadrangle. Write the measure of each angle
on the angle.
3.
Find the sum of the angles. Write the sum of the angles on your quadrangle.
4.
Cut out your quadrangle and try to make a tessellation by tracing your
quadrangle repeatedly. Draw your tessellation in the space provided below or
on the back of this page.
(Hint:: Label your angles A, B, C, and D so you can be sure that all
four angles meet at each vertex.)
5.
Repeat Steps 1–4 for a different convex quadrangle. Try to tessellate your
second quadrangle. Draw your tessellation on the back of this page.
6.
Do both of your quadrangles tessellate?
7.
Do you think that all convex q
quadrangles
g
will tessellate?
Yes
Yes
Sample
p answer: The sum of
the measures of the four angles
g
of any
y
convex quadrangle
q
g equals
q
360°, so
the quadrangle
q
g can be repeated
p
about
ap
point, with a vertex of each
quadrangle
q
g meeting
g at that point
p
with
no gaps
g p and no overlapping.
pp g
p
Whyy or whyy not?
g
SMALL-GROUP
ACTIVITY
An Angle Investigation
py g
EXTRA PRACTICE
Time
Do all convex quadrangles tessellate? (A convex quadrangle is one in which
all vertices are pushed outward.) To find out, do the following:
PARTNER
ACTIVITY
ENRICHMENT
Date
Math Masters, p. 335
329-349_EMCS_B_G6_MM_U10_576981.indd 335
2/26/11 3:18 PM
Lesson 10 2
889