Regular Tessellations
Objective To explore side and angle relationships in regular
ttessellations and compare and classify quadrangles.
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Teaching the Lesson
Key Concepts and Skills
• Use angle relationships to determine angle
measures. [Geometry Goal 1]
• Describe the properties of regular polygons. [Geometry Goal 2]
• Compare and classify quadrangles. [Geometry Goal 2]
• Identify, describe, and create tessellations. [Geometry Goal 3]
Key Activities
Students are introduced to the history and
concept of tessellations; they explore regular
tessellations and decide which regular
polygons tessellate and which ones do not,
based on the sum of the angle measures
around a single point. They compare and
classify quadrangles.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Angle Tangle
Student Reference Book, p. 296
Math Masters, p. 444
Geometry Template (or straightedge
and protractor)
Students practice estimating and
measuring angles.
Math Boxes 3 8
Math Journal 1, p. 84
Students practice and maintain skills
through Math Box problems.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Making Tessellations with Pattern Blocks
pattern blocks or Geometry Template
Students explore tessellations using a
concrete model.
ENRICHMENT
Naming Tessellations
Math Masters, p. 91
Students create regular tessellations
to explore the naming conventions
for tessellations.
Study Link 3 8
Math Masters, p. 90
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters,
page 414). [Geometry Goal 2]
Key Vocabulary
regular polygon tessellation regular
tessellation tessellate tessellation vertex
Materials
Math Journal 1, pp. 82 and 83
Student Reference Book, pp. 146, 160,
and 161
Study Link 37
Math Masters, pp. 87A, 89 and 414
Geometry Template scissors
Advance Preparation
For Part 1, make copies of Math Masters, page 89 and place them near the Math Message. The Study Link
for this lesson asks students to collect examples for a Tessellation Museum. Prepare a space in your classroom for this display. For
a mathematics and literacy connection, obtain a copy of A Cloak for the Dreamer by Aileen Friedman (Scholastic Inc., 1995).
Teacher’s Reference Manual, Grades 4–6 pp. 201–206
194
Unit 3
Geometry Explorations and the American Tour
Mathematical Practices
SMP1, SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Getting Started
Content Standards
5.NBT.2, 5.NF.5a, 5.G.3, 5.G.4
Mental Math and Reflexes
Math Message
Use your established slate procedures. Pose the following problems. Students
respond by writing their magnitude estimate—placing their solution in the
1,000s; 10,000s; 100,000s; or 1,000,000s.
Follow the directions on
Math Masters, page 89.
18 ∗ 200 1,000s
300 ∗ 12 1,000s
200 ∗ 19 1,000s
5 ∗ 48,000 100,000s
13 ∗ 500,000 1,000,000s
60 ∗ 5,000 100,000s
28 ∗ 3,020 10,000s
39 ∗ 5,130 100,000s
4,000 ∗ 527 1,000,000s
Study Link 3 7
Follow-Up
Partners compare answers
and resolve any differences.
Then they exchange and solve
Odd Shape Out problems.
1 Teaching the Lesson
▶ Math Message Follow-Up
(Math Masters, p. 89)
WHOLE-CLASS
DISCUSSION
ELL
Allow time for students to finish cutting out the polygons on Math
Masters, page 89. Review the names of the polygons. Ask students
to verify that each polygon’s sides are the same length and their
angle measures are equal. Tell students that such polygons are
called regular polygons. To support English language learners,
write regular polygons on the board along with some examples.
These cut-out polygons may be discarded at the end of this lesson.
▶ Exploring Tessellations
WHOLE-CLASS
DISCUSSION
(Student Reference Book, pp. 160 and 161)
As a class read and discuss pages 160 and 161 of the Student
Reference Book. Highlight the following points.
A tessellation is an arrangement of repeated, closed shapes
that cover a surface so no shapes overlap and no gaps exist
between shapes. (See margin.)
A tessellation
Teaching Master
Name
LESSON
38
䉬
Date
Time
Regular Polygons
Cut along the dashed lines. Fold the page like this along the solid lines.
Cut out the polygons. You will be cutting out four of each shape at once.
Some tessellations repeat only one shape. Others combine two
or more shapes.
A tessellation with shapes that are congruent regular polygons
is called a regular tessellation.
89
Math Masters, p. 89
Lesson 3 8
195
Student Page
Date
LESSON
38
䉬
Time
Adjusting the Activity
Regular Tessellations
1.
A regular polygon is a polygon in which all sides are the same length and all angles have the
same measure. Circle the regular polygons below.
2.
In the table below, write the name of each regular polygon under its picture. Then, using the
polygons that you cut out from Activity Sheet 3, decide whether each polygon can be used to
create a regular tessellation. Record your answers in the middle column. In the last column,
use your Geometry Template to draw examples showing how the polygons tessellate or don’t
tessellate. Record any gaps or overlaps.
Tessellates?
(yes or no)
Polygon
Draw an Example
ELL
Have students identify tessellations that they see around them—in
ceiling tiles, floor tiles, carpet designs, clothing designs, and so on. Ask which of
these tessellations use only one shape and whether any are made with regular
polygons. For example, the floor or ceiling might be tiled with squares. Remind
students that in a regular polygon, all the sides are the same length and all the
angles have the same measure.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Yes
▶ Exploring Regular Tessellations
triangle
(Math Journal 1, pp. 82 and 83; Math Masters, p. 89)
Yes
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VING
VIN
V
IIN
NG
square
No
gap
pentagon
Have students use the regular polygons that they cut from Math
Masters, page 89 to help them complete the tables on journal
pages 82 and 83 and answer the questions on journal page 83.
For each of the given regular polygons, partners must decide
whether the polygon can be used to create a regular tessellation.
Ask students to use their Geometry Templates to draw an
example of each tessellation. For polygons that do not tessellate,
the drawing should show an overlap or a gap in the design.
Math Journal 1, p. 82
Ask volunteers to share their results from the journal page with
the class. Then survey the class: Which regular polygons will
tessellate and which ones will not? The triangle, square, and
hexagon tessellate; the pentagon and octagon do not.
Student Page
Date
LESSON
38
䉬
Time
Regular Tessellations
Polygon
Tessellates?
(yes or no)
continued
Draw an Example
Yes
hexagon
overlap
No
octagon
3.
Ask students to examine their drawings on journal page 82. What
true statements can they make about the angles in the drawings?
For the triangle and the square, the sum of the measures of the
angles around a single point is 360°; for the pentagon, the sum of
the angle measures around a single point is not 360°. Note that
the point where vertices meet in a tessellation is called the
tessellation vertex. Ask students what true statements they can
make about their drawings on journal page 83. For the hexagon,
the sum of the angle measures around a tessellation vertex is
360°; for the octagon, the sum of the angle measures is not 360°.
Conclude the discussion by asking students to use what they
know about the total number of degrees in a circle and the
measure of the angles in regular polygons to determine which
regular polygons will tessellate and which ones will not. A regular
polygon can be tessellated if a multiple of the measure of its
angles equals 360°. Each angle in a regular pentagon is 108°. No
multiple of 108° equals 360°, so there will be overlaps or gaps if
pentagons are arranged around a point.
Which of the polygons can be used to create regular tessellations?
Triangles, squares, and hexagons
4.
Three pentagons
leave a gap, and 4 pentagons create an overlap. For regular
polygons that have 7 or more sides, 2 shapes leave a gap,
and 3 shapes create an overlap.
Explain how you know that these are the only ones.
Math Journal 1, p. 83
196
Unit 3
Geometry Explorations and the American Tour
Teaching Master
▶ Quadrangles
INDEPENDENT
ACTIVITY
Name
LESSON
37
Date
Time
Classifying Quadrangles
NOTE For this activity, students will need the completed Math Masters,
Quadrangles
Distribute Math Masters, page 87A. Remind students of the work
they did when they classified quadrangles in Lesson 3-7. Draw a
tree diagram, like the one below, on the board. Ask volunteers to
draw shapes on the board for each category.
trapezoids
not parallelograms
page 87A from Lesson 3-7.
rhombuses
Quadrangles
kites
other
(Math Masters, p. 87A)
Sample shapes:
not parallelograms
rectangles
other
squares
kites
rectangles
trapezoids
parallelograms
parallelograms
rhombuses
Math Masters, p. 87A
EM3cuG5MM_U03_067-101.indd 87A
1/12/11 3:12 PM
squares
Record the following statements on the board. Ask students to
identify each statement as true or false. Students defend their
thinking using logical arguments. Refer students to their tree
diagrams as a resource, if necessary.
All squares are parallelograms. true
All rhombuses are rectangles. false
A kite is a rhombus. false
All quadrangles are parallelograms. false
Trapezoids are not parallelograms. true
All rhombuses are parallelograms. true
Record the following sentences on the board, along with the words
always, sometimes, and never. Ask students to make each sentence
true by using the word always, sometimes, or never.
Squares are
rectangles. always
Rhombuses are
rectangles. sometimes
Trapezoids are
rectangles. never
A kite is
Rectangles are
a parallelogram. never
squares. sometimes
Ask students to explain why each of the above statements is
always, sometimes, or never true. Encourage students to use
what they know about the properties of quadrangles in their
explanations and to refer to the tree diagram as needed.
Lesson 3 8 196A
Ask students to use their knowledge of the relationships among
quadrangles to generate three statements similar to those
discussed in the second group of statements on page 196A. Have
students record the statements on an Exit Slip, Math Masters, page
414. The statements should include one of each type of response—
always, sometimes, or never—to make it true. Have students share
their statements with a partner. Circulate and assist.
Ongoing Assessment:
Recognizing Student Achievement
Math Masters
Page 414
Use the statements on Exit Slip, Math Masters, page 414 to assess students’
abilities to classify quadrangles according to a hierarchy of properties. Students
are making adequate progress if their statements include a basic understanding
of the classification of quadrangles. Some students may demonstrate a more
sophisticated understanding. For example, a square is always a rhombus, a
rectangle, and a parallelogram.
[Geometry Goal 2]
Write the three statements listed below on the board. To extend
students’ understanding of the properties of quadrangles, ask them
to work with a partner to write each of the statements on a sheet
of paper, inserting the names of quadrangles in the blanks and
then indicating if the statement is true or false. An example has
been given for each.
If it is a
, then it is also a
.
Example: If it is a rectangle, then it is also a parallelogram. true
are
.
All
Example: All trapezoids are parallelograms. false
are
.
Some
Example: Some squares are rhombuses. false
Circulate and assist.
196B Unit 3
Geometry Explorations and the American Tour
Student Page
Games
2 Ongoing Learning & Practice
Angle Tangle
Materials 䊐 1 protractor
䊐 1 straightedge
Estimating and measuring angle size
80
100
90
100
80
110
70
12
60 0
13
50 0
0
15 0
3
20
160
160
20
Object of the game To estimate angle sizes
accurately and have the lower total score.
10
170
170
10
180
0
0
180
Directions
In each round:
(Student Reference Book, p. 296; Math Masters, p. 444)
70
0
60 0 11
12
3
15 0
0
Skill
50 0
13
0
14 0
4
▶ Playing Angle Tangle
PARTNER
ACTIVITY
2
4
14 0
0
䊐 several blank sheets of paper
Players
1. Player 1 uses a straightedge to draw an angle on a
sheet of paper.
Students practice estimating angle measures and measuring
angles with a protractor by playing Angle Tangle. Students draw
angles and record their answers and points on the Angle Tangle
Record Sheet.
▶ Math Boxes 3 8
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 3-10. The skill in Problem 6
previews Unit 4 content.
Writing/Reasoning Have students write a response to the
following: Blaire wrote the following true statement based
on the questions for Problem 6: 450 is 90 times as great as
5. Write similar statements for the question “How many 5s are in
35,000?” Sample answer: 35,000 is 7,000 times as great as 5.
3. Player 1 measures the angle with a protractor.
Players agree on the measure.
4. Player 2’s score is the difference between the estimate
and the actual measure of the angle. (The difference
will be 0 or a positive number.)
5. Players trade roles and repeat Steps 1–4.
Players add their scores at the end of five rounds.
The player with the lower total score wins the game.
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 84)
▶ Study Link 3 8
2. Player 2 estimates the degree measure of the angle.
INDEPENDENT
ACTIVITY
(Math Masters, p. 90)
Home Connection Students collect tessellations that they
can bring to class. Students can draw tessellations that
they find if they cannot cut them out.
Estimate
Player 1
Actual
Score
Estimate
Player 2
Actual
Score
Round 1
Round 2
120°
75°
108°
86°
12
11
50°
85°
37°
87°
Round 3
40°
44°
4
15°
19°
13
2
4
Round 4
60°
69°
9
40°
56°
16
Round 5
135°
123°
12
150°
141°
9
Total score
48
44
Player 2 has the lower total score. Player 2 wins the game.
Student Reference Book, p. 296
NOTE Several math supply catalogs offer
paper pattern blocks. These are already cut
to the correct shapes and colors. They only
need to be separated and glued down.
Tessellations can also be explored using
computer software or online sites such as
the Tessellation Creator provided by the
National Council of Teachers of Mathematics
at http://illuminations.nctm.org/ActivityDetail.
aspx?ID=202.
Game Master
Name
Date
Time
1 2
4 3
Angle Tangle Record Sheet
Estimated
measure
Actual
measure
1
_______°
_______°
2
°
_______
°
_______
3
°
_______
°
_______
4
°
_______
°
_______
5
_______°
_______°
Round
Angle
Score
Total Score
Math Masters, p. 444
Lesson 3 8
197
Student Page
Date
Time
LESSON
3 8
Circle the name(s) of the shape(s) that
could be partially hidden behind the wall.
1.
3 Differentiation Options
Math Boxes
䉬
rectangle
pentagon
2.
Which triangles are congruent?
a and c
a.
b.
c.
d.
▶ Making Tessellations with
e.
rhombus
155
143 146
Trace an isosceles triangle using your
Geometry Template.
3.
4.
Art Link To explore tessellations using a concrete model,
have students create tessellating patterns using pattern
blocks.
28°
88°
P
207
144
Solve.
6.
1
If four counters are ᎏ
2 , then what is
one whole?
5
3
How many 60s in 5,400? 90
How many 5s in 35,000? 7,000
How many 80s in 5,600? 70
How many 700s in 2,100?
8 counters
9 counters
Solve.
How many 90s in 450?
1
If 3 counters are ᎏ
3 , then what is
one whole?
Pattern Blocks
A
M
74
15–30 Min
64⬚
What is the measure of angle A?
Sample answers:
5.
INDEPENDENT
ACTIVITY
READINESS
They should trace their patterns onto a piece of paper, either by
tracing around the blocks or by using the Geometry Template.
Suggest that students color their patterns in a way that
emphasizes repeating elements.
INDEPENDENT
ACTIVITY
21–22
ENRICHMENT
▶ Naming Tessellations
Math Journal 1, p. 84
15–30 Min
(Math Masters, p. 91)
To explore naming conventions for tessellations, have students
create and label tessellations using Geometry Template polygons.
Students focus on the vertex points of tessellations and the
number of polygons that are arranged around a tessellation
vertex.
Teaching Master
Study Link Master
Name
Date
STUDY LINK
38
䉬
Name
Time
Date
LESSON
Tessellation Museum
38
䉬
A tessellation is an arrangement of repeated, closed shapes that completely
covers a surface, without overlaps or gaps. Sometimes only one shape is used
in a tessellation. Sometimes two or more shapes are used.
160 161
Time
Naming Tessellations
Regular tessellations are named by giving the number of sides in each
polygon around a vertex point. A vertex point of a tessellation is a point
where vertices of the shapes meet.
160
tessellation vertex
4.4.4.4
1.
Collect tessellations. Look in newspapers and magazines. Ask people at home
to help you find examples.
2.
Ask an adult whether you may cut out the tessellations. Tape your
tessellations onto this page in the space below.
3.
If you can’t find tessellations in newspapers or magazines, look around your
home at furniture, wallpaper, tablecloths, or clothing. In the space below,
sketch the tessellations you find.
For example, the name of the rectangular tessellation above is 4.4.4.4. There are four
numbers in the name, so there are four polygons around each vertex. Each of those numbers
tells the number of sides in each of the polygons around a vertex point. The numbers are
separated by periods. There are four 4-sided polygons around each vertex point.
Look at the tessellation below.
Choose a vertex.
6
3
What is the name of this regular tessellation? 3.3.3.3.3.3
Why? Because there are six 3-sided polygons
around each vertex
1. How many shapes meet at the vertex point?
2. How many sides does each polygon have?
3. a.
b.
4. Make a tessellation for each regular polygon on your geometry template. Use
the back of this page if necessary. Name each regular tessellation.
Sample answers:
Practice
10,246
29,712
1,467
5 R4
4.
1,987 ⫹ 6,213 ⫹ 2,046 ⫽
5.
4,615 ⫺ 3,148 ⫽
6.
3,714 º 8 ⫽
7.
39 / 7 →
4.4.4.4
Math Masters, p. 90
198
Unit 3
Geometry Explorations and the American Tour
3.3.3.3.3.3
Math Masters, p. 91
6.6.6
87A
squares
rhombuses
other
kites
37
trapezoids
not parallelograms
LESSON
Date
Copyright © Wright Group/McGraw-Hill
rectangles
parallelograms
Quadrangles
Name
Time
Classifying Quadrangles