Regular Tessellations - Login...Key Activities Students are introduced to the history and concept of tessellations; they explore regular tessellations and decide which regular polygons

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194 Unit 3 Geometry Explorations and the American Tour

Advance PreparationFor 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

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 ActivitiesStudents 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.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). [Geometry Goal 2]

Key Vocabularyregular polygon � tessellation � regular

tessellation � tessellate � tessellation vertex

MaterialsMath Journal 1, pp. 82 and 83

Student Reference Book, pp. 146, 160,

and 161

Study Link 3�7

Math Masters, pp. 87A, 89 and 414

Geometry Template � scissors

Playing Angle TangleStudent Reference Book, p. 296

Math Masters, p. 444

Geometry Template (or straightedge

and protractor)

Students practice estimating and

measuring angles.

Math Boxes 3�8Math Journal 1, p. 84

Students practice and maintain skills

through Math Box problems.

Study Link 3�8Math Masters, p. 90

Students practice and maintain skills

through Study Link activities.

READINESS

Making Tessellations with Pattern Blockspattern blocks or Geometry Template

Students explore tessellations using a

concrete model.

ENRICHMENTNaming TessellationsMath Masters, p. 91

Students create regular tessellations

to explore the naming conventions

for tessellations.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

�� Regular TessellationsObjective To explore side and angle relationships in regular

tessellations and compare and classify quadrangles.t

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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LESSON

3�8

Name 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.

89Math Masters, p. 89

Teaching Master

Lesson 3�8 195

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Masters, p. 89)

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-CLASSDISCUSSION

(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.)

� 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.

ELL

Math Message Follow the directions on Math Masters, page 89.

Study Link 3�7 Follow-Up Partners compare answers and resolve any differences. Then they exchange and solve Odd Shape Out problems.

Mental Math and Reflexes 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.

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

A tessellation

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Regular TessellationsLESSON

3�8

Date Time

1. A regular polygon is a polygon in which all sides are the same length and all angles have thesame measure. Circle the regular polygons below.

2. In the table below, write the name of each regular polygon under its picture. Then, using thepolygons that you cut out from Activity Sheet 3, decide whether each polygon can be used tocreate 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’ttessellate. Record any gaps or overlaps.

PolygonTessellates?

Draw an Example(yes or no)

triangle

square

pentagon

Yes

Yes

No gap

Math Journal 1, p. 82

Student Page

Regular Tessellations continuedLESSON

3�8

Date Time

3. Which of the polygons can be used to create regular tessellations?

4. Explain how you know that these are the only ones. Three pentagons

Polygon Tessellates? Draw an Example(yes or no)

hexagon

octagon

Yes

No

overlap

Triangles, squares, and hexagons

leave a gap, and 4 pentagons create an overlap. For regularpolygons that have 7 or more sides, 2 shapes leave a gap,and 3 shapes create an overlap.


Student Page

Adjusting the Activity


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

▶ Exploring Regular Tessellations PARTNER ACTIVITY

(Math Journal 1, pp. 82 and 83; Math Masters, p. 89)

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.

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.

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.

ELL

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Name Date Time

LESSON

3�7 Classifying Quadrangles

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Math Masters, p. 87A

Teaching Master

▶ Quadrangles

INDEPENDENT ACTIVITY

(Math Masters, p. 87A)

NOTE For this activity, students will need the completed Math Masters,

page 87A from Lesson 3-7.

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.

Quadrangles

parallelograms

squares

not parallelograms

rectangles rhombuses

kites othertrapezoids

Sample shapes:

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 a parallelogram. never

� Rectangles are 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

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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: Math Masters

Page 414 �Recognizing Student Achievement

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

� All are . Example: All trapezoids are parallelograms. false

� Some are . Example: Some squares are rhombuses. false

Circulate and assist.

196B Unit 3 Geometry Explorations and the American Tour

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Angle Tangle

Materials � 1 protractor � 1 straightedge� several blank sheets of paper

Players 2Skill Estimating and measuring angle sizeObject of the game To estimate angle sizes accurately and have the lower total score. DirectionsIn each round: 1. Player 1 uses a straightedge to draw an angle on a

sheet of paper.

2. Player 2 estimates the degree measure of the angle.

3. Player 1 measures the angle with a protractor. Players agree on the measure.

4. Player 2’s score is the difference between the estimateand the actual measure of the angle. (The differencewill 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.

Games

Player 1 Player 2

Estimate Actual Score Estimate Actual Score

Round 1 120° 108° 12 50° 37° 13Round 2 75° 86° 11 85° 87° 2Round 3 40° 44° 4 15° 19° 4Round 4 60° 69° 9 40° 56° 16Round 5 135° 123° 12 150° 141° 9

Total score 48 44

Player 2 has the lower total score. Player 2 wins the game.

908070

60

50

40

3020

10

100 110 120130

140150

160170

1800

100110

120

130

140

150

160

170

80 7060

50

4030

2010

0

180

Student Reference Book, p. 296

Student Page

Name Date Time

Angle Tangle Record Sheet 132

4

Round Angle Estimated Actual Scoremeasure measure

1 _______° _______°

2 _______° _______°

3 _______° _______°

4 _______° _______°

5 _______° _______°

Total Score

Math Masters, p. 444

Game Master

Lesson 3�8 197

2 Ongoing Learning & Practice

▶ Playing Angle Tangle PARTNER ACTIVITY

(Student Reference Book, p. 296; Math Masters, p. 444)

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


(Math Journal 1, p. 84)

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.

▶ Study Link 3�8



Home Connection Students collect tessellations that they can bring to class. Students can draw tessellations that they find if they cannot cut them out.

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.

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Math Boxes LESSON

3 � 8

Date Time

5. Solve.

If four counters are , then what is

one whole?

If 3 counters are , then what is

one whole? 9 counters

1�3

8 counters

1�2

6. Solve.

How many 90s in 450?

How many 700s in 2,100?



How many 80s in 5,600? 70

903

5

1. Circle the name(s) of the shape(s) thatcould be partially hidden behind the wall.

rectangle pentagon rhombus

3. Trace an isosceles triangle using yourGeometry Template.

Sample answers: 4. What is the measure of angle A? 64�

207

21–22

144

143 146

74

2. Which triangles are congruent?

a. b.

c. d.

e.

a and c

P

A

88°28°M

155

7,000


Student Page

STUDY LINK

3�8 Tessellation Museum

160 161

Name Date Time

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.

1. Collect tessellations. Look in newspapers and magazines. Ask people at hometo help you find examples.

2. Ask an adult whether you may cut out the tessellations. Tape yourtessellations onto this page in the space below.

3. If you can’t find tessellations in newspapers or magazines, look around yourhome at furniture, wallpaper, tablecloths, or clothing. In the space below,sketch the tessellations you find.

4. 1,987 � 6,213 � 2,046 � 5. 4,615 � 3,148 �

6. 3,714 º 8 � 7. 39 / 7 → 5 R429,7121,46710,246

Practice

Math Masters, p. 90

Study Link Master Teaching Master

LESSON

3�8

Name Date 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.

4.4.4.4

For example, the name of the rectangular tessellation above is 4.4.4.4. There are fournumbers in the name, so there are four polygons around each vertex. Each of those numberstells the number of sides in each of the polygons around a vertex point. The numbers areseparated by periods. There are four 4-sided polygons around each vertex point.

Look at the tessellation below.

Choose a vertex.

1. How many shapes meet at the vertex point?

2. How many sides does each polygon have?

3. a. What is the name of this regular tessellation?

b. Why? Because there are six 3-sided polygonsaround each vertex

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:

3.3.3.3.3.336

tessellation vertex

4.4.4.4 3.3.3.3.3.3 6.6.6

160

Math Masters, p. 91


3 Differentiation Options

READINESS


▶ Making Tessellations with 15–30 Min

Pattern BlocksArt Link To explore tessellations using a concrete model, have students create tessellating patterns using pattern blocks.

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.

ENRICHMENT


▶ Naming Tessellations 15–30 Min


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.


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Documents

Regular Tessellations - Login...Key Activities Students are introduced to the history and concept of tessellations; they explore regular tessellations and decide which regular polygons