Investigating Symmetryinvestigating at the Center of the Figure Center the At

Home , Geometry, Reflection (mathematics), Reflection symmetry, Rotational symmetry, Symmetry (geometry)

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5/14/14 7:33 PM edition. hardcover student find thislessoninthe Turn to thesepagesto edition. hardcover student find thislessoninthe Turn to thesepagesto Date PAGES 8996 symmetry Line of Resource Locker esn 4 Lesson 5/14/14 7:32PM 102 translation, reflection, translation,

n lines of symmetry Investigating SymmetryInvestigating at the center of the figure center the at

What are the three rigid motions explained explained rigid the three motions are What does a rigid motion What in this module? When you are testing a figure to see if it has see has to a figure if it testing are you When of P, the center is where symmetry, rotational EXPLORE 1 1 EXPLORE EXPLORE 2 2 EXPLORE INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING INTEGRATE TECHNOLOGY INTEGRATE STRATEGIES QUESTIONING rotation; shape and size rotation; transformation preserve? rotation? Identifying Line Symmetry Identifying Have students use geometry software or cut out out cut use or geometry software students Have the symmetry polygons. examine regular to figures of make to reasoning them use inductive have Then symmetry of lines a of the number about conjectures regular has. n-gon Identifying Rotational Symmetry Rotational Identifying using of cons and discuss the to pros students Ask of properties investigate to geometry software recognize Be students sure symmetry. and rotations it making of the advantage has software such that of the angle as (such parameters change easy to they observe so can that the effects the of rotation) changes. DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 12:35 PM

Lesson Lesson 4 Angle of rotational Angle symmetry: 72° one A circle 102 120° zero 240° , the star coincides with with coincides = 72°, the star

5 360° ____ 120°, two Identifying Rotational Symmetry Rotational Identifying line has the figure If a rotation. as such a rigid motion, undergoes A figure Math Background Math figure with your pencil. Rotate the traced figure counterclockwise until it coincides again again coincides it until the traced counterclockwise figure pencil. Rotate your with figure beneath. the originalwith figure symmetry has. figure each

Discussion What figure has an infinite number of lines of symmetry? of lines of number infinite an has figure What What do you have to know about any segments and angles in a figure to to in a figure angles and segments any about know to have you do What symmetry? line decide whether has the figure symmetry, does the image of the figure have line symmetry as well? Give an example. an symmetry line Give well? as have the figure of does the image symmetry, Yes. The line of symmetry also undergoes the rigid motion. For example, if the L-shape in if the L-shape example, line of symmetry motion. For the rigid The also undergoes Yes. the same way. the line of symmetry is rotated V-shape, a into D is rotated Step Pairs of segments in the figure must have the same length and pairs of angles must have length and pairs of angles must have the same must have in the figure of segments Pairs the other half when the with will coincide figure one half of the so that the same measure, a line of symmetry. across is folded figure the original the traced against figure of the center Hold paper. tracing onto the figure Trace of lines of number the total tell and figure each on if any, symmetry, of the lines Draw

2 Explore PROFESSIONAL DEVELOPMENT PROFESSIONAL A wallpaper pattern is a planar repeating pattern. Mathematicians classify Mathematicians pattern. repeating a planar is pattern A wallpaper only example, for based the symmetries they on exhibit, patterns wallpaper symmetry; reflection symmetry; and translation translation all these or plus its identifying be can classified by pattern Every wallpaper symmetry. rotational is, That classifications. 17 different precisely are there symmetries. Surprisingly, types. 17 basic of one to be can reduced a plane covers that pattern repeating any of in a number applications far-reaching fact had has mathematical unusual This chemistry crystallography. and fields, including How many lines many How of symmetry? Figure

. If °. If 288 72°, 144°, 216°, and are the star for rotation of angles The itself. has the figure the original, exactly match to rotates the figure of a copy symmetry. rotational By how many degrees did you rotate the figure? the figure? rotate you did degrees many how By rotation? of all the angles are What  Module Module 2 1 onto the figure symmetry maps rotational has A figure if a rotation symmetry 0° but than rotational angle of greater The is , which itself. a maps that rotation of angle the smallest 180°, is equal to or than less itself. onto figure that symmetry a fractional 360°. Notice is part of rotational of angle An every rotates star time the 5-pointed 3. 2. Reflect 1. 

GE_MNLESE385795_U1M02L4.indd 102

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© Houghton Mifflin Harcourt Publishing Company • no symmetry • rotational but not symmetry line symmetry • two lines of symmetry • one (vertical) line of symmetry • one (horizontal) line of symmetry ofwith each type symmetry: example, capital the letters F, G,and J. Have students name aletter of alphabet the Have students identifycommon shapes that do not Small Group Activity COLLABORATIVE LEARNING xli A figure may have line symmetry, rotational symmetry, Explain1 of types symmetry, both or no symmetry. 5. 4. Reflect  oue 2 Module  Example 1 each figure Determine whether has rotational symmetry. If so, angles the identifyall of have an angle of rotational of 60°?If not, symmetry what is angle the of rotational symmetry? Discussion What figure is mapped onto itself by arotation of any angle? intersection willonlymaponeofthelinesontointersection theotherline. No, for theangleofrotational thefigure symmetry is180°.Arotation of60 ° aboutthe Step 1 Step rotation less than 360°. less than360° Angles ofrotation Figure Describe the symmetry of figure. Draw each of lines the symmetry, name symmetry the Describe of for byfigure. the Look findingline the symmetry matching Begin halves of Notice that lines the intersect at center the of figure. the topthe over half bottom the half. Draw one for line of each fold. symmetry figure.the For example, you could fold over half left the right the half, and fold the angles of angles the rotation, or hasboth. figure ifthe both Afigure is formed by line land line m,which intersect at an angle figure the of 60°.Does F GJKLPQR Describing Symmetries 90°, H IOX 180°,

270° M T U VW Y A M T B CDE 103 none

N SZ A circle have line symmetry, for 180° esn 4 Lesson 3/20/14 12:35PM 104

to a figure with line with a figure to Investigating SymmetryInvestigating . If you identify the line of symmetry of identify the line the on you . If CONNECT VOCABULARY CONNECT Connect a reflection theidea of symmetry y-axis the x- or on line that superimpose and figure, symmetry of then the line plane, a coordinate on see can the you reflection, and of the line becomes side. either on preimage and image DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 12:35 PM

© Houghton Mifflin Harcourt Publishing Company is Lesson 4 120°, 240° that are are that is added to form this plural. In In this plural. form added to is to the end of a noun to form form to a noun of the end to es . In English, some nouns are are nouns some English, . In Angles of rotation: rotation: of Angles and 270°. 180°, and

Angle of rotational Angle symmetry: 90° Angles of rotation: 90°, 180°, 270° 90°, 180°, rotation: of Angles 104 3

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120° center until it matches the original figure. It rotates around the circle by a by the circle around rotates It the original figure. matches it until center rotation, of find 360° to the angle by . Multiply fraction of angles find other to this angle of multiples . Find is which rotation. of symmetry: of lines of Number Look for lines of symmetry. One line divides the figure into left and right right left and into the figure divides line One symmetry. of lines Look for the begin at that lines similar draw Then the figure. on this line Draw halves. the figure. vertices of other its about rotating the figure Think of symmetry. rotational lookfor Now Next, look for rotational symmetry. Think of the figure rotated about about rotated the figure Think of symmetry. rotational look for Next, rotational of angle The original position. its matches it until center its symmetry is this figure of Now look for other lines of symmetry. The two diagonals also describe diagonals two The symmetry. of lines other look for Now symmetry. of 4 lines of a total has figure The halves. matching Number of lines of symmetry: of lines of 4 Number The other angles of rotation for the figure are the multiples of 90° of the multiples are the figure for rotation of angles other The °. So the angles of rotation are 90°, are rotation 360°. So of than less the angles is added to form the plural. form added to is

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LANGUAGE SUPPORT LANGUAGE In English and in Spanish, we usually add –s or add usually –es we in Spanish, and English In triangles, example, for the plural, points, figures vertex the noun of form plural The convention. that follow don’t and irregular Connect Vocabulary Connect vertices a c and z becomes in z. The end that words with happens thing the same Spanish, then –

Module 2

GE_MNLESE385795_U1M02L4.indd 104

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” so that theimagecoincides withthepreimage. angle greater than0°andlessorequalto 180° if thefigure canberotated aboutapoint by an with thepreimage. Afigure has rotational symmetry reflected across alinesothat theimagecoincides ifthefigure canbe figure hasline symmetry order 2.Afive-pointed star is of order 4. a square is of order 3and an equilateral is triangle of and repeat process. the picturethe that matches clues. the switch They roles has 4lines of symmetry.” The second student picks rotation is 90degrees; it has 4sides and 4angles; and rotational and line symmetry. symmetry Its angle of gives oral clues figure suchhas both as, “This draws acard, such as picture the of asquare, and students and one is down. face set The first student ofset pictures is up placedface pair the of between or neither. Students get of two sets each figure. One that have line symmetry, rotational symmetry, both, order the 360° is called Focus onReasoning Focus onCommunication esn 2 .4 Lesson 105 MP.2 MP.3 PRACTICES INTEGRATE MATHEMATICAL PRACTICES INTEGRATE MATHEMATICAL SUMMARIZE THE LESSON SUMMARIZE THE ELABORATE Give students pictures of figures or objects The number of angles of rotation less than line symmetry or rotational line symmetry symmetry? How do you determine afigure whether has of rotational the symmetry, so A CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough “File info” GE_MNLESE385795_U1M02L4 105

© Houghton Mifflin Harcourt Publishing Company original mandala.original examples andfind describe of rotational ineach. Have symmetry create them an orBring inbooks suggest websites that show examples of mandalas. Have students Modeling DIFFERENTIATE INSTRUCTION 8. 6. ofangles rotation, or hasboth. figure ifthe both for figure. Draw each of lines the symmetry, of symmetry type the name the Describe Module 2 Module 11. 10. Your Turn rotational symmetry, themappingisby rotation. motion. Inlinesymmetry, thefigure ismappedonto itselfby andin reflection, show how ofsymmetry afigure bemappedonto types can Both itselfby arigid symmetry? Angles of rotation: Number of lines of symmetry: Types of symmetry: Figure Angles of rotation: Number of lines of symmetry: Types of symmetry: Figure Check-InEssential Question How How alike? are of two the types symmetry are different? they produces theoriginalfigure. rotating around itscenter. The figure ifarotation hasrotational ofat symmetry most180° figure into mirror-image halves. To identify rotational symmetry, thinkofthefigure Possible answer: To identify linesymmetry, lookfor that alineofreflection dividesthe Elaborate KLNPR ABCD D S R Q P 72°, D A 90°, 144°, line, rotational J K rotational N 180°, 216° and288°

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Figure Figure Angles of rotation: Number of lines of symmetry: Types of symmetry: Angles of rotation: Number of lines of symmetry: Types of symmetry: l E TUVW EFGHI W T U V none none none line H F 1 0 Lesson 4 Lesson

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Evaluate: Homework and Practice EVALUATE

• Online Homework Draw all the lines of symmetry for the figure, and give the number of lines of • Hints and Help symmetry. If the figure has no line symmetry, write zero. • Extra Practice 1. 2. 3.

1 8 1 Lines of symmetry: Lines of symmetry: Lines of symmetry: ASSIGNMENT GUIDE Concepts and Skills Practice For the figures that have rotational symmetry, list the angles of rotation less than 360°. For figures without rotational symmetry, write “no rotational symmetry.” Explore 1 Exercises 1–3 4. 5. 6. Identifying Line Symmetry Explore 2 Exercises 4–6 Identifying Rotational Symmetry Example 1 Exercises 7–16 45°, 90°, 135°, 180°, Angles of Angles of Angles of Describing Symmetries no rotational symmetry 225°, 270°, 315° no rotational symmetry rotation: rotation: rotation:

In the tile design shown, identify whether the pattern has line symmetry, rotational symmetry, both line and rotational symmetry, or no symmetry. INTEGRATE MATHEMATICAL 7. 8. PRACTICES

(t) MP.1 It is also possible to define a symmetry based ©costall/Shutterstock on translations. A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. Tiled floors may be ; (b) examples of this. ©Cbenjasuwan/Shutterstock rotational symmetry both line and rotational symmetry

For figure ABCDEF shown here, identify the image after each transformation A _ described. For example, a reflection across AD has an image of figure F B AFEDCB. In the figure, all the sides are the same length and all the angles are E C the same measure. D _ 9. Reflection across CF 10. rotation of 240 clockwise, or 11. reflection across the line that ° _ 120 counterclockwise connects the midpoint of BC ° _ and the midpoint of EF EDCBAF CDEFAB DCBAFE Figure Figure Figure

Module 2 106 Lesson 4

COMMON GE_MNLESE385795_U1M02L4.inddExercise 106 Depth of Knowledge (D.O.K.) CORE Mathematical Practices 16/10/14 3:49 PM

1 1 Recall of Information MP.4 Modeling 2 2 Skills/Concepts MP.4 Modeling 3–4 1 Recall of Information MP.4 Modeling 5 2 Skills/Concepts MP.4 Modeling 6 1 Recall of Information MP.4 Modeling 7–8 2 Skills/Concepts MP.4 Modeling

Investigating Symmetry 106 CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” esn 2 .4 Lesson 107 and for fourth, the 90°. should map logo the onto itself; for third, the 120°; logothe onto itself. For second, the arotation of 180° first logo,the there should no rotations be that map Have students create four different, simple logos. For to all find tothem reread directionsare the ifthey to asked see or oneline of angle symmetry of rotation. Remind Some students may stop have they when found one JOURNAL AVOID COMMON ERRORS lines of symmetry or linesall of symmetry angles of rotation. CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” GE_MNLESE385795_U1M02L4.indd 107

15. 12. characteristics. sketchIn provided, an space example the ofgiven the with afigure Module 2 Module 16. H.O.T. c. b. a. explain why. each figure shown. Identify student’s the work as correct or incorrect. If incorrect, Communicate Mathematical Ideas line the and rotationalDescribe figure. inthis symmetry 180 symmetry: angle of rotationalno line symmetry; Explain theError Use concepts inyour of answer. symmetry Incorrect; thefigure hasnolinesofsymmetry. rotational symmetry. haveBoth two perpendicularlinesofsymmetry, andbothhave 180° 90° angleofrotationalfour symmetry: linesofsymmetry; the remaining pairsofopposite vertices. Incorrect; thefigure hasthree more linesofsymmetry, eachconnecting Incorrect; thetwo diagonalsare notlinesofsymmetry.

Focus onHigherOrder Thinking 3 2 3 2 2 Depth ofKnowledge (D.O.K.) Skills/Concepts Skills/Concepts Skills/Concepts Strategic/Thinking Strategic Thinking Possible answers are shown. °

A student was asked to draw on of all lines the of symmetry

How is arectangle similar to an ellipse?

107 MP.5 MP.6 MP.4 MP.4 MP.2 13. COMMON CORE UsingTools Precision Modeling Modeling Reasoning symmetry no rotationalone line of symmetry; Mathematical Practices Lesson 4 Lesson 3/20/14 12:34PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-B;CA-B

Lesson Performance Task AVOID COMMON ERRORS In evaluating rectangular shapes for symmetries, students sometimes identify the diagonals as lines of symmetry. Unless a rectangle is a square, its diagonals are not lines of symmetry.

INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Students should recognize that while single elements of their designs might exhibit symmetries, those symmetries might not extend to the entire design. Use symmetry to design a work of art. Begin by drawing one simple geometric figure, such as a triangle, square, or rectangle, on a piece of construction paper. Then add other lines or two- For example, in a design composed of a square and dimensional shapes to the figure. Next, make identical copies of the figure, and then arrange them in a symmetric pattern. four isosceles triangles, each of those shapes contains

Evaluate the symmetry of the work of art you created. Rotate it to identify an angle of rotational lines of symmetry. None of those lines, however, is a symmetry. Compare the line symmetry of the original figure with the line symmetry of the line of symmetry for the entire design. finished work.

Answers will vary. Students’ responses should identify all lines of symmetry (horizontal, vertical, and diagonal) as well as all angles of rotational symmetry (90°, 180°, and 270°). © Houghton Mifflin Harcourt Publishing Company Company Publishing Harcourt Mifflin © Houghton

Module 2 108 Lesson 4

EXTENSION ACTIVITY

GE_MNLESE385795_U1M02L4 108 5/14/14 4:16 PM The flags of many nations have rotational symmetry or line symmetry. The flags of a few nations, such as Jamaica, have both. Research the flags of the nations of the world to find examples of symmetry. If you wish, disregard color and concentrate only on the designs of the flags. Draw or print out examples of designs you find especially interesting or attractive.

Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Investigating Symmetry 108