Technology Assessment Management System

Journal page 281, Problem 1See the iTLG.

See the Web site on page 821.

816 Unit 10 Reflections and Symmetry

Teaching the Lesson materials

Key ActivitiesStudents read about frieze patterns in which a design is repeatedly reflected, rotated, or translated to produce a pattern. They complete frieze patterns and create their own.

Key Concepts and Skills• Identify and draw congruent figures. [Geometry Goal 2]• Identify, describe, and sketch reflections, rotations, and translations.

[Geometry Goal 3]• Extend, describe, and create geometric patterns. [Patterns, Functions, and Algebra Goal 1]

Key Vocabularyfrieze pattern • reflection (flip) • translation (slide) • rotation (turn)

Ongoing Assessment: Recognizing Student Achievement Use journal page 281.[Geometry Goal 3]

Ongoing Learning & Practice materialsStudents play Polygon Pair-Up to practice identifying properties of polygons.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use patternblocks to create andcontinue geometricpatterns.

Students explore different arrangements of four straws.

Students use technology toexplore tessella-tions.

Students cut out atemplate and createfrieze patterns.

� Teaching Master (Math Master, p. 319)� Teaching Aid Masters (Math Masters,

pp. 389 and 437)� Geometry Template; straws; pattern

blocks; computer with Internet access;index cards; scissors

EXTRA PRACTICEENRICHMENTENRICHMENTREADINESS

3

� Math Journal 2, p. 282� Student Reference Book, p. 258� Study Link Master (Math Masters,

p. 318)� Polygon Pair-Up Polygon Cards and

Property Cards (Math Masters,pp. 496 and 497)

2

� Math Journal 2, p. 281� Student Reference Book, p. 108� Study Link 10�4 � Teaching Aid Master (Math Masters,

p. 403; optional)� overhead or regular pattern blocks

(optional)� 1 transparent mirror per partnership� straightedge� slate

1

Objective To guide the application of reflections, rotations,

and translations.

� Math Message Follow-Up(Student Reference Book, p. 108)

Ask students to describe what they notice about each of the friezepatterns on Student Reference Book, page 108. Have students indicate thumbs-up if they made a similar observation. The discussion should include the following points:

� Each pair of horses in the first frieze pattern is the same sizeand shape, but they face in opposite directions.

� The line of reflection for each pair of horses is a vertical linerunning through the center of the flower.

� Each elephant and horse in the second frieze pattern is thesame size and shape and faces in the same direction.

� Each of the flowers in the third frieze pattern is the same sizeand shape but is turned in a different direction.

Tell students that in this lesson they will explore examples offrieze patterns to see how designs are created by repeatedlymoving figures in three different ways: reflections, translations,and rotations.

� Introducing Frieze Patterns(Student Reference Book, p. 108)

The examples of frieze patterns on Student Reference Book, page 108 provide an initial exposure to “rigid motions,” in whichobjects retain their shape as they are moved in various ways.

WHOLE-CLASS

DISCUSSION

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

Lesson 10�5 817

Getting Started

Study Link 10�4 Follow-UpHave partnerships compare their Venn diagrams. Students should note that the letter O has an infinite number of lines of symmetry if it is printed as a circle.

Mental Math and ReflexesPose mental subtraction problems. Suggestions:

78 – 58 � 20

49 – 37 � 12

51 – 20 � 31

84 – 63 � 21

94 – 85 � 9

77 – 58 � 19

46 – 27 � 19

97 – 59 � 38

150 – 42 � 108

283 – 179 � 104

402 – 203 � 199

731 – 446 � 285

Math MessageRead page 108 in your Student Reference Book.Be prepared to describe what you notice about the three frieze patterns.

The Mayan peopledecorated their buildingswith painted sculpture,wood carvings, and stonemosaics. The decorationswere usually arranged inwide friezes.

Geometry and Constructions

Frieze Patterns

A frieze pattern is a design made of shapes that are in line.Frieze patterns are often found on the walls of buildings, on the borders of rugs and tiled floors, and on clothing.

In many frieze patterns, the same design is reflected over andover. For example, the following frieze pattern was used todecorate a sash worn by a Mazahua woman from San FelipeSantiago pueblo in the state of New Mexico. The strange-looking animals reflected in the frieze are probably meant to be horses.

Some frieze patterns are made by repeating (translating) thesame design instead of reflecting it. These patterns look as ifthey were made by sliding the design along the strip. Anexample of such a frieze pattern is the elephant and horsedesign below that was found on a woman’s sarong from Sumba,Indonesia. The elephants and horses repeated in the frieze areall facing in the same direction.

The following frieze pattern is similar to one painted on thefront page of a Koran in Egypt about 600 years ago. (The Koranis the sacred book of Islam.) The pattern is more complicatedthan the two above. It was created with a combination ofreflections, rotations, and translations.

Student Reference Book, p. 108

Student Page

818 Unit 10 Reflections and Symmetry

281

Frieze Patterns LESSON

10 � 5

Date Time

108

�1. Extend the following frieze patterns. Use a straightedge and your transparent mirror

to help you. Then write if you used a reflection, rotation, or translation of the original shape to continue the pattern.

a.

b.

c.

2. Create your own frieze pattern. Make a design in the first box. Then repeat the design, using reflections, slides, rotations, or a combination of moves. When you have finished, you may want to color or shade your frieze pattern.

3. Explain how you created your pattern in Problem 2.Answers vary.

rotation

translation

reflection

Math Journal 2, p. 281

Student Page

NOTE You may wish to use the less formalterms: flip (reflection), turn (rotation), andslide (translation). To support English language learners, demonstrate the mathematical meanings with gestures.

Adjusting the Activity

Through work in the preceding lessons, students should befamiliar with reflections. (The top frieze on Student ReferenceBook, page 108 was created by reflecting the horse repeatedlyacross vertical lines of reflection.) The second frieze is an exampleof another rigid motion—a translation, or slide, in which a shapeis moved without being turned or flipped. To support Englishlanguage learners, discuss the everyday and mathematical uses ofthe word translation.

Students should observe a difference between the first and second friezes: In the first frieze, the design is repeated so that alternating designs face in opposite directions. In the secondfrieze, the designs all face in the same direction.

The third frieze on the Student Reference Book page combinesslides, flips, and turns in a complex design.

Overhead or regular pattern blocks also can be used to illustraterigid motions. For example, the pattern below was made by arotation of each figure 90° clockwise, followed by a translation of the figure to the right.

� Drawing Frieze Patterns(Math Journal 2, p. 281)

Art Link Students complete three frieze patterns on journalpage 281 and design one of their own.

ELL

Have students sketch the original design for each frieze pattern on centimeter grid paper (Math Masters, page 403). They can use a transparentmirror to check their reflections for Problem 1a and slide and rotate the paper to check their patterns for Problems 1b and 1c.

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

Ongoing Assessment:Recognizing Student Achievement

Use journal page 281, Problem 1 to assess students’ ability to identify andsketch an example of a reflection and identify examples of translations androtations. Students are making adequate progress if they are able to continue the pattern in Problem 1a and identify the transformations in Problems 1a–1c.Some students may be able to sketch the translations and rotations in Problems 1b and 1c.

[Geometry Goal 3]

Journal

page 281

Problem 1�

INDEPENDENT

ACTIVITY

A pattern made by rotations and translations

� Playing Polygon Pair-Up(Student Reference Book, p. 258; Math Masters, pp. 496 and 497)

Students play Polygon Pair-Up to practice identifying properties of polygons. See Lesson 1-6 for additional information.

� Math Boxes 10�5(Math Journal 2, p. 282)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-2. The skill in Problem 5 previews Unit 11 content.

Writing/Reasoning Have students write a response to the following: Calculate the mean number of hours Juliababysat last week and explain your strategy. Sample answer:

I added the hours she babysat each day (4 � 2�12� � 1 � 1�

12� � 3 � 12)

and divided by 5 for the number of days. 12 / 5 � 2�25� hours, or

2 hours 24 minutes.

� Study Link 10�5(Math Masters, p. 318)

Home Connection Students extend geometric patternsand create patterns of their own.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

2 Ongoing Learning & Practice

282

Math Boxes LESSON

10 � 5

Date Time

2. Insert the decimal point in each quotient.

a. 74.8 � 4 � 1 8 7b. 6 9 � 34.5 � 5

c. 88.5 � 3 � 2 9 5d. 2 4 2 � 193.6 � 8•

•

•

•

3. Julia babysat for the family next door. Below are the hours she worked.

Monday: 4 hours

Tuesday: 2�12� hours

Wednesday: 1 hour

Thursday: 1�12� hours

Friday: 3 hours

Use this data to create a bar graph.

1. Complete the table with equivalent names.

22 23

76

Julia’s Babysitting Hours

Days of the Week

Num

ber o

f Hou

rs

M Tu W Th F012

1121

212231234

4. Write the ordered pair for each pointplotted on the coordinate grid.

A ( , )

B ( , )

C ( , )

D ( , )

E ( , )

4035315324

5. Write five names for �73.

a.

b.

c.

d.

e.�36 �

12� � (�36 �

12�)

10 � (�83)�75.5 � 2.5

�75 � 2�70 � (�3)

144 60

5 D

CE

BA4

3210

0 1 2 3 4 5

Sample answers

0.31�11000� 10%

25%0.050.25

�1500�

31%

�146�

0.10

5%

�13010�

61 62

Fraction Decimal Percent

Math Journal 2, p. 282

Student Page

Lesson 10�5 819

STUDY LINK

10�5 Geometric Patterns

106–108

Name Date Time

1. Continue each pattern. Then tell if you continued the pattern by using areflection, rotation, or translation of the original design.

a.

b.

c.

2. Make up your own pattern. Answers vary.

rotation

translation

reflection

3. 50% of $25.00 � 4. 25% of $10.00 �

5. � 40% of $150.00 6. � 20% of $250.00$50.00$60.00$2.50$12.50

Practice

Math Masters, p. 318

Study Link Master

820 Unit 10 Reflections and Symmetry

Name Date Time

Dot Paper

Math Masters, p. 437

Teaching Aid Master

� Exploring Geometric Patterns(Math Masters, p. 389)

Art Link To explore geometric patterns using a concretemodel, have students create and continue patterns using

pattern blocks. One student begins a pattern. The other studentsin the group take turns continuing the pattern.

A group drawing can be made of some of the patterns by tracingthe blocks or using the Geometry Template. On an Exit Slip, havestudents describe how they created their pattern using words suchas slide, flip, or turn.

� Exploring Arrangements of Four Straws(Math Masters, p. 437)

To apply students’ understanding of congruence, reflections, androtations, have them investigate 16 possible ways to arrange fourstraws. Explain the following:

� Each straw must connect to the end of at least one other straw,and the straws must connect to form either a straight angle ora right angle.

yes no

� A flipped or rotated figure cannot be counted as a different figure.

Have students record their figures on Math Masters, page 437 and explain the strategy they used to find them.

15–30 Min

INDEPENDENT

ACTIVITYENRICHMENT

15–30 Min

SMALL-GROUP

ACTIVITYREADINESS

3 Differentiation Options

EM07TLG2_G4_U10_L05.qxd 1/19/07 1:39 PM Page 820

LESSON

10�5

Name Date Time

Making Frieze Patterns

1. Use an index card as a template for making frieze patterns.

a. Trim your index card to make a 3-inch by 3-inch square.

b. Draw a simple design in the middle of the square.

c. Cut out your design. If you need to cut through the edge of the index card,then use tape to repair the cut.

2. Make a frieze pattern with your template.

a. Draw a long line on a large sheet of paper.

b. Put your template at the left end of the line.

c. Trace the shape of the design you cut out. Make a mark on the line at theright edge of the template.

d. Move your template to the right along the line. Line up the left side of thetemplate with the mark you made on the line.

e. Repeat Steps c and d. To make more complicated patterns, give your template a turn or a flip every time you move it.

108

Math Masters, p. 319

Teaching Master

Lesson 10�5 821

The three regular tessellations

Links to the Future

� Exploring TessellationsTechnology Link To apply students’ understanding of reflections, rotations, and translations, have them

use technology to explore tessellations. See the Web site at http://nlvm.usu.edu/en/nav/frames_asid_163_g_2_t_3.html?open�activities.

Discuss the following points:

� A tessellation is an arrangement of repeated, closed shapes thatcover a surface so that no shapes overlap and there are no gapsbetween the shapes.

� Some tessellations repeat only one basic shape. Others combinetwo or more basic shapes. A regular tessellation is a tessellationmade up of only one kind of regular polygon.

� In a tessellation, the basic shapes are reflected, rotated, ortranslated to fill the surface.

Ask students which of the regular polygons shown on the Web sitecan be used to create a regular tessellation. There are only threeregular tessellations—equilateral triangle, square, and hexagon.

Students will further explore regular tessellations in Fifth Grade EverydayMathematics. Naming notation, semi-regular tessellations, and Escher-typetranslation tessellations are introduced in Sixth Grade Everyday Mathematics.

� Creating Frieze Patterns(Math Masters, p. 319)

Art Link To practice reflections, rotations, and translations,have students make frieze patterns by following thedirections on Math Masters, page 319. Ask them todescribe any reflections, translations, or rotations they use.

15–30 Min

INDEPENDENT

ACTIVITYEXTRA PRACTICE

overlapgap

15–30 Min

INDEPENDENT

ACTIVITYENRICHMENT