Sequences of Transformations

Home , Translation (geometry)

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© Houghton Mifflin Harcourt Publishing Company   Find image the for each sequence of transformations. image of first transformation the as preimage the for next the transformation. plane.the Transformations applied can be one after other the inasequence where you the use A transformation is that afunction takespoints on plane the and maps to them other points on Explore Essential Question: 3.1 Name oue 3 Module   Rotate Rotate Reflect Reflect Consider the relationship Consider the △DEF between Using software, geometry draw and atriangle labelthe Using software, geometry draw and atriangle labelthe Make aconjecture regarding asingle rotation that map △A″BC. will to △ABC transformation that maps △DEF halfway to △D around ofjandkby theintersection theanglemadebetween thelinesrotates itabout A rotation of75° label the imagelabel the as △A″BC △A′BC vertices vertices vertices Check your conjecture, what and you describe did. A rotation withcenter at ofj theintersection tosoftware rotate seethat △ABC75°,Ican thisimagecoincides with△A″BC. triangle andtriangle labelit P. and k. jand them label image as △D″EF △D′EF 30°around△ABC point Pand image labelthe as △DEF ′. Then rotate △A′BC . Then reflect △D′EF Then′. reflect A, D, Sequences of TransformationsSequences B, and C.Then draw apoint outside the E, and F.Then draw two intersecting across line jand image labelthe as What happenswhenyou applymore thanonetransformation to afigure? ″ Combining Rotations orReflections E ″. Sketch your result. ″ ( F because 30+45=75 because ″, sorotate itby twice that angleto see△DEFmappedto △D ″. Sketch your result. ′ 45°around point P and label the ′ across the line kand label

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6/19/14 12:54 AM edition. hardcover student find thislessoninthe Turn to thesepagesto edition. hardcover student find thislessoninthe Turn to thesepagesto Date D A″ ″ E F B″ C″ B ″ j ″ C P E ″ ″ F′ . By usingthe PAGES 103112 E′ F C′ ″ A′ . Rotating △DEF C D′ F″ B′ ″ E ″ F A D″ k ″ Resource B . E″ Locker esn 1 Lesson 6/19/14 12:54AM 116 For reflections in For 1 Sequences of Transformations How can you use geometry software to check check use to geometry you software can How transformations? your EXPLAIN EXPLAIN EXPLORE EXPLORE AVOID COMMON ERRORS COMMON AVOID INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING Combining Combining Rigid Transformations the original figure transform may students Some get to the image first transforming of twice instead that Note the third. get the to second and the second, A → A' with transformations two when performing A' is and the preimage is A the transformation, first as A' → A", A' transformation the second In the image. the image. A" is and the preimage is Combining Reflections Combining the completing of the option have Students in the either activity reflections combining online. book or parallel lines, use the measuring features to see if all to use the measuring features lines, parallel in the same the same distance move points rotate reflections in intersecting lines, direction. For the see if the images are to figure the preimage and shape. same size 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 5:27 PM

. ″ ℓ ″. C v ⇀

″ A ℓ C C ″ B ′ along . Check your ″. Check your ′ C B B A A ′ △A′B′C ν ν ⇀ A ⇀ ′ B to △D″E″F Translate Translate Label this image △A″B″C Label this image 2 Step 116

′. v ⇀

ℓ C ′ C B A ′ ν ⇀ A ′ B Combining Combining Rigid Transformations , so two reflections in || lines with △D″E″F″, so two image coincides The it . △DEF by a Make intersecting. j and of lines make k parallel instead C, but Step Repeat . Label the image A′B′C △ ℓ. Label line the image across transformations. of the combination given △ABC after the of image Draw Draw the image of △ABC a reflection after of the image 1 Draw of three rotations about the same center can be described by a single rotation rotation a single can be described by about the same center rotations of three sequence by the sum of the angles of rotation. the sum of the angles by A . I marked a vector from D to D″ and from a vector of △DEF. I marked a translation △D″E″F″ looks like translated in a translation. result If a figure is rotated and then the image is rotated about the same center, a single rotation rotation a single about the same center, the image is rotated and then is rotated If a figure result. the same will have the angles of rotation the sum of by Make a conjecture about what single transformation will describe a sequence of three rotations will rotations describe three of sequence a transformation single what about a conjecture Make about the same center. Repeat Step A using other angle measures. Make a conjecture about what single transformation transformation single what about conjecture a Make measures. angle other using A Step Repeat center. the same about will describe rotations two of a sequence Discussion conjecture and describe what you did. you conjecture and describe what conjecture about what single transformation will now map △DEF map will now transformation single conjecture about what Step along ℓ then translation line Reflection over

PROFESSIONAL DEVELOPMENT PROFESSIONAL Students have worked with individual transformations and should now be able to to be able now should and transformations individual with worked have Students they this lesson, In rotations. and reflections, describeidentify and translations, of sequences include may and these transformations of more or two combine the outcome predict visualize and to be able They must transformations. nonrigid other consider as well as transformation, one than more performing of the lesson Throughout final the same image. produce that transformations the methods for and transformation each of recall the properties they must them. drawing Math Background Math Example 1

 describe to the Explain 1 use transformation a single can you sometimes that saw you the Explore, In rigid of sequences will apply you Now transformations. two of a sequence applying of result be described transformation. a single cannot by that transformations 3. 2. 1. Reflect Module Module 3

GE_MNLESE385795_U1M03L1.indd 116

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© Houghton Mifflin Harcourt Publishing Company their predictionstheir more accurate. students have done for this figures, ask several to them brainstorm ways to make transformationsperform the and check results the against predictions. their After sketch aprediction image. of final the Then have software geometry to use them ofseries transformations. Instruct to them plot points the on graph paper and to drawing multiple transformations. Give students coordinates the of afigure and a software allows studentsGeometry to on focus predictions their rather than on Small Group Activity COLLABORATIVE LEARNING

B Module 3 Module Draw image of the after triangle the given the combination of transformations. 6. 5. 4.

Your Turn Reflect

image backto thepreimage. Possible answer: Inthiscase, reversing theorder ofthetransformations willtake thefinal above lineℓinstead ofbelow it. Possible answer: Yes, first, ifIreflect thenrotate, andthentranslate, thefinalimageis but they donotchangethesize orshape. Yes. transformations Rigid move thefigure intheplaneandmay changetheorientation, 7. For Part B, asequence of transformations describe that △A″BCback take to preimage. the will performing transformations the inPart Binadifferent order. order the Does inwhich you apply transformations the make adifference? Test your conjecture by In what ways transformations do rigid change preimage? the Are images the you drew for each example same the sizeand shape as given the preimage? Apply translation the to △A′BC Apply rotation. the image the Label △A′BC across line ℓ 180° rotation around point translation P,then along Apply to reflection △A″BC the point across ℓ Reflection B ′′ P A A C ′′ ′′ P then 90°rotationthen around B C ℓ C B ′ ′ . Label the image the ″. Label △A‴BC . Label the image the ′. Label △A″BC A ′ ′. 117

⇀ v , then reflection reflection then ,

8. ″. ‴. point Translation along Translation F G ′′′ then translation Pthen along E G E ′ ′′′ F ′′ E G F ′ A C ′′′ B ′

⇀ P v P E ′′

F then 180°rotationthen around ′ C B A ′ ′ G ⇀ v ′′ ⇀ ν ⇀ u A A ″

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Lesson 1 Lesson B C B C ″ ′′ ′′ ′ ″ ′ ℓ 6/19/14 12:54AM 118

to rigid , the Latin word , the Latin word

2 Either the size or the shape of the or the shape Either the size No. The polygon will have the will have polygon The No.

derives from rigidus from derives is used to represent a type of used is represent to by comparing the size and shape of of shape the size and comparing by Sequences of Transformations If you perform a sequence of nonrigid nonrigid of a sequence perform you If will the type polygon a polygon, of on motions How would you describe the image of a figure a figure describe of the image you would How nonrigid of a sequence after . Help students understand how nonrigid how understand students . Help Relate nonrigid Relate transformation EXPLAIN EXPLAIN CONNECT VOCABULARY CONNECT INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES STRATEGIES QUESTIONING Explain. change? transformations? for stiff The word rigid word The MP.1 transformation a different is that image an gives that transformation that out Point figure. a preimage of shape size and/or in the or be can in the plane the transformation transformation a nonrigid that and plane, coordinate combined of sequence in any becan included transformations. Combining Combining Transformations Nonrigid Connections on Math Focus transformation a that out Point figures. image the original and a the size of not but the preserves shape dilation does not vertical or stretch while a horizontal figure, a figure. of the shape the size or preserveeither same number of vertices, so it will be the same same number of vertices, is regular, If figure the original polygon. general of a an image give motions may the nonrigid be may image of a square The polygon. non-regular example. for a parallelogram, original figure changed, although it is possible that although it is possible that changed, figure original of in a figure results transformation a subsequent and shape. size the original DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 5/14/14 4:57 PM

© Houghton Mifflin Harcourt Publishing Company ′ x C x 8 Lesson Lesson 1 2 ′ ′′ C D y 6 A A ′ 0 A 6 4 2 4 6 D - - 4 B 2 ′′ ′ ′ B - A B 2 B A 4 C - y ′ B 0 6 ′′ 2 8 6 4 C - ′′′ ′′′ B A 2 ′′ ′′ 8 B - A - 4 ′′′ 10 - D ′′ - D 6 ′ 12 - C - 8 ′′′ C - ′′ C 118

) - 2 △A′B′C′ . Possible answer: You could dilate the figure by a factor by the figure dilate could You answer: . Possible + 1, y

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Apply the transformationApply image image -axis. Label this image A″B″C Labely-axis. this image A″B″C of the translation Apply image image A′B′C the reflection of Apply A″'B″'C this image ] a [horizontal/vertical is transformation first The . of a factor by stretch . Label the image the stretch. Apply a factor by a dilation is transformation second The of The first transformation is a dilation by a by a dilation is transformation first The factor of y y 1 2 _ x, x, ( ( of Dilate the figure by a factor of a factor by of the figure Dilate 3. 2. 2. 1. 1.

-axis and then vertically stretch the figure the figure then vertically stretch and the y-axis reflect to across a figure asked is A student transformation one write Then 2. Describe of a factor the effect the coordinates. on by one. into these transformations two combines that notation coordinate using If you dilated a figure by a factor of 2, what transformation could you use to return the figure the figure return use to you could transformation 2, what of a factor by a figure dilated you If 2 units, right it then translated 2 and of a factor by a figure dilated you If its preimage? to back preimage. its to back the figure return to transformations of a sequence write The The of 2.

DIFFERENTIATE INSTRUCTION DIFFERENTIATE Have students graph any three points on a coordinate plane and connect them to them to connect and plane a coordinate on points three any graph students Have this triangle. on transformations two perform to students Ask a triangle. form rules transformations. the perform same instruct use to Then them the to algebra those with they algebraically they found the coordinates compare should Students and the preimage them study have Then transformation. the physical with found to transformation usedone decide whether they to the final have could image the that rules show to use them the to algebraic ask so, If result. the same obtain original transformations. the two to equivalent is transformation single Multiple Representations Example 2

Module Module 3  2 Explain 10. 9. Reflect 

GE_MNLESE385795_U1M03L1 118

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© Houghton Mifflin Harcourt Publishing Company another of set images. explanation under images. the Students change roles and repeat sequence the with and tell it whether or is rigid nonrigid, and why. The first student writes the nonrigid transformation. The second student transformation the should describe preimage and transformed image and ask it whether is an example of or arigid Have students work inpairs. Have first student the show agraph partner the of a Communicate Math LANGUAGE SUPPORT oue 3 Module The final resultsame the atriangle Theshape final be and will sizeas △LMNinQuadrant IV. It has of Predict third effect the the transformation: across Areflection again x-axis the will of Predict second effect the transformation: the Since is triangle inQuadrant the II, of Predict first transformation: effect the the A  Explain3 11. Draw image of the plane inthe after figure the given the combination of transformations. Your Turn Example 3

( across y-axis, the andacross reflected then x-axis. the △LMN is translated along vector the reflected 〈-2,3〉, x, been rotatedbeen 180°about origin and the from is farther axes the than preimage. the equivalent of rotating figure the 180°about origin. the change orientation the and move into triangle the Quadrant IV. The two reflections are the Quadrant I. acrossa reflection change y-axis the will orientation the and move into triangle the Quadrant II. fromit x-and the further y-axes. It remain will in istriangle inQuadrant translation II,the move will figure 2units left and up 3units. Since given the translation along vector move the 〈-2,3〉will the y C - ) ′′′

→ 6 Predict the result the Predict of applying of transformations sequence the to the given figure. ( - x 4 - 1,y B - ′′′ A 2 ′′′ Predicting of theEffect Transformations - - - - 6 4 2 2 4 6 1 0 )

A → y B ′ ′ ( A C B 2 3x, ′ A B ′′ y C ′′ )

→ ( -x, 6 C ′′ -y x )

119 12.

( x, y - )

6 → B - ′′ - 6 ( _ 2 3 4 L

x, - M -2y - 4 2 A ′′ - - - - )

→ 6 4 2 2 4 6 2 0 - - - B B ( y ′ 6 4 2 4 6 N x 0 - 5,y C y ′′ 2 A A 2 ′ + 4 4 C ) 4

6 C ′ 6 esn 1 Lesson x x 3/20/14 5:26PM 120 Sequences of Transformations DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 5:26 PM

6 6 6 Lesson 1 J H 4 4 I J 2 2 K G H K y y y V 0 0 0 6 4 2 4 6 6 4 2 2 4 6 6 4 2 2 4 6 ------U 2 2 - - 4 4 4 T - - - 6 6 6 - - -

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a square in Quadrant 4 with side lengths twice as long as the as long with side lengths twice 4 in Quadrant a square will move the rectangle 5 units right will move and then translated along the vector 〈2, 1〉. the vector along then translated , and ) is rotated 90° clockwise about the origin and the and origin about clockwise 90° rotated is y 〈5, 4〉 x,

2 3 -axis and translated translated and the y-axis reflectedGHJK is across __

( HIJK will move the stretched triangle 2 units right triangle 2 units right the stretched will move 〉 →

) y 2, 1 -axis will move the rectangle the y-axis will move reflection across The the left of the y-axis to Due the right to of it. from the symmetry have it will appear to of the rectangle, along translating Then, left 6 units. been translated the vector and up 4 units. This will bring the rectangle fully into bring the rectangle will fully into This and up 4 units. will be a rectangle final result that The I. Quadrant has that as the preimage is the same shape and size the closer to I, sit on the x-axis in Quadrant to moved y-axis than the preimage.

x, side lengths of the original. The image is furtherThe preimage. the origin than the from side lengths of the original. 〈 by a factorby the side lengths of the of 2 will double It will also be further the origin than square. from the preimage. and 1 unit up, which will move the triangle closer to to the triangle closer which will move and 1 unit up, on the x-axis and another the origin with one vertex will not be the same final image The the y-axis. across as the preimage. shape or size A horizontal stretch will pull points U and T away will pull points stretch A horizontal making the y-axis, the triangle longer in the left-from along the translation The to-right direction. rotation will map it to Quadrant IV. Due its to IV. Quadrant will map it to rotation but been translated, have will appear to it symmetry, y-axis. the is to the x-axis than it to will be closer Square

The final result will be finalThe result

the effect the transformation: first Predict of the the transformation: effect second Predict of then dilated by a factor of 2, which maps maps 2, which of a factor by then dilated ( is horizontally stretched by a factor of of a factor by stretched horizontally △TUV is Rectangle Rectangle 〈5, 4〉. the vector along

Your Turn Your

14. Predict the result of applying the sequence of transformations to the given figure. the given to the sequence transformations of applying of Predict the result 13.

B Module 3

GE_MNLESE385795_U1M03L1.indd 120

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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" the size orshapeofthefigure haschanged. from theoriginal figure, itsorientation, andwhether and inwhatitwillbefrom direction theorigin or which quadrant(s) thefinalimagewillbein,how far represented by x-axis followed by y-axis inthe is areflection represented by example, x-axis inthe is represented areflection by nonrigid motions coordinate inthe plane. For algebraic patterns to represent of and aseries rigid Focus onPatterns esn 3 .1 Lesson 121 ( MP.8 transformation? original figure. image musthave thesamesize andshapeasthe preserves thesize andshapeofafigure, sothefinal the transformations. write aseriesofrelated transformations to undo be undonedirectly, butstudents may beableto different rigid motionsornonrigid motionscannot sequence. ispossiblethat asequence It using sometimes beundoneby reversing theorder ofthe transformations usingthesamerigid motioncan PRACTICES INTEGRATE MATHEMATICAL SUMMARIZE THE LESSON SUMMARIZE THE QUESTIONING STRATEGIES ELABORATE x, y ) → Discuss with Discuss students how to record and use predicting result the of more than one What features can you when describe undone?can be What of types sequences of transformations motions always rigid? Is atransformation from asequence of rigid ( x, –y ) ( ( , while a reflection in the y-axis inthe is areflection ,while

x, y x, y Sample answer:Sample You canpredict ) ) → → A sequence of ( ( –x, y –x, y ) Yes. Eachrigid motion .So, inthe areflection ) . CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L1.indd 121

© Houghton Mifflin Harcourt Publishing Company 16. 15. Module 3 Module 17. You addextra can transformations to findadditionalsequences. you went down inatranslation, soyou take can apreimage backonto itselfinmany ways. over aline. You always can justasfaryou gobackleft went right, orupasmany timesas you andyou started, dothat can asmany timesasyou want. You always can back reflect An infinite number. With rotations you justneedto go360°andyou willbebackwhere sequence ofsequence transformations where order the matter. does asequence image. of final transformations the affect Describe where order the not does matter. a Describe preimage back onto itself? Explain your reasoning. as preimage? the Explain your reasoning. Is there asequence of arotation and adilation that result will inan image that is same the sizeand position Discussion Check-InEssential Question 4 followed by arotation of90°. of 4willhaveby afactor adifferent finalimagethanthesamefigure dilated of by a factor thefinalimage,affects for example arotation of90°around avertex followed by adilation When asequence includesamixofdifferent oftransformations, types theorder usually counterclockwise rotations. Any sequence oftranslations alsobedoneinany can order. added together to make onerotation, even ifthey are acombination ofclockwiseand Possible answer: Asequence ofany numberofrotations aboutthesamepoint be can Yes, arotation of360°andadilation of1willwork. Elaborate

How many different sequences of transformations rigid do you you think canto a find take

In asequence of transformations, order the of transformations the can 121 Lesson 1 Lesson 3/20/14 5:26PM 122 Exercises 1–4 Exercises 5–6 Exercises 7–12 Exercises Practice igid Transformations Transformations igid Sequences of Transformations Students can use geometry software to do a do use to geometry can software Students Concepts and Skills Concepts Explore or Reflections Rotations Combining Example 1 R Combining Example 2 Nonrigid Combining Transformations Example 3 the Effect of Predicting Transformations EVALUATE EVALUATE INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES ASSIGNMENT GUIDE ASSIGNMENT Focus Focus on Technology MP.5 sequence of transformations or to check a sequence a sequence check to or transformations of sequence use the to students Remind transformations. of rigid of a sequence verifyto that features measuring a figure. of shape the preserves size and motions 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 5:26 PM

© Houghton Mifflin Harcourt Publishing Company x x ′′′ C 8 Lesson Lesson 1 8 ′′′ x B 6 ′′′ B 6 6 ′′ ′ C ′′′ C • Online Homework • Hints and Help • Extra Practice ′′′ 4 A C ′′ ′ 4 ′′′ B 4 B ′ A A ′ 2 C 2 2 ′′ ′ y A A ′ y B y 0 6 4 2 2 4 6 A 0 8 6 4 2 - - - 0 ′ 6 4 2 2 4 6 A C 2 A A - - - ′ C 2 - Mathematical Practices Mathematical C C 2 - B - ′ 4 B ′′ B B Precision Reasoning Reasoning Reasoning Reasoning 4 ′′ A - 90 degrees clockwise about the △ABC about clockwise 90 degrees 4 A - ′′ CORE - C 6 COMMON MP.6 MP.2 MP.2 MP.2 MP.2 6 - 6 ′′ ′′ Rotate the-axis. x then reflect and over origin - B - B 8 ′′ 8 2. - C - 122 x 6 ′′ B ′′ 4 C ′ B ′′ A ′ 2 C ′ A y 0 2 4 6 6 4 2 A - - - C Skills/Concepts Recall of Information 2 Skills/Concepts Skills/Concepts Skills/Concepts - Depth of Knowledge (D.O.K.) Depth of Knowledge 2 2 2 2 1 B 4 - , rotate 90 degrees 90 degrees △ABC by 〈4, 4〉, rotate -axis, translate by by translate the x-axis, △ABC over -axis and then translate by by then translate and the y-axis △ABC over 6 , and rotate 180 degrees around the around 180 degrees rotate -1〉, and - y-axis. 14 19 -3〉. 1–8 9–13 15–18 Evaluate: Homework and Practice and Homework Evaluate: 〈-3, origin. Reflect Translate Translate reflect A, and over around counterclockwise the 〈2, Reflect Exercise

4. 3. Module Module 3 1. after the sequence given label and the final △ABC after Draw of image of transformations.

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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" esn 3 .1 Lesson 123 restthe of class, the explaining each step. transform. Then have each group present its poster to motions. Give each group adifferent sequence to coordinatethe plane and using rigid nonrigid both showing how toasequence of find transformations in Have students work groups insmall to make aposter get adifferent result order ifthe is reversed. planein the it and x-axis.will inthe reflect They then correct order by asking to them rotate 90° atriangle importance of doing transformations the inthe transformations wrong inthe order. Emphasize the Some students may perform acombination of For example: proceed with sequence the ofthey transformations. help remember them last to the image use figure as properties inasequence of transformations. This can track of types the transformations and their Suggest that students agraphic use organizer to keep GRAPHIC ORGANIZERS SMALL GROUP ACTIVITY AVOID COMMON ERRORS Property 1 Property 1 Property 1 Property Transformation 3 Transformation 2 Transformation 1 Transformation ↓ ↓ Property 2 Property 2 Property 2 Property CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L1 123

© Houghton Mifflin Harcourt Publishing Company Exercise shape, orientation. The finalimageisthesamesize with will mapthefigure mostlyinto Quadrant III, and changetheorientation. The second reflection Predict the result the Predict of applying of transformations sequence the to given the figure. 5. image ofDraw △ABCafter final the and label of transformations. given the sequence Module 3 Module reflection ofthey-axisandC’the left closerto theorigin. The threedown units, oneunitandleft mappingA’to Possible answer: 7. 21–23 20 ( A’’’ inQuadrant IV, andagainchangethe x, across y-axis. the acrossreflected x-axis, the and reflected then is translated△ABC along vector the 〈-3, and orientation asthepreimage. y - C - ) ′′

6 4 → first willmapA’’ - (

2 3 Depth ofKnowledge (D.O.K.) - x, Skills/Concepts 4 2 Strategic Thinking B _

A C 3 1 - - - ′′′

′ ′′′ y A - 6 4 2 2 4 6 8 ) The translation moves thefigure ′′ 0

A 2 → ′′ - - - y A B (

6 4 2 2 4 6 8 B ′′′ B B ′ 0 -2x, A ′′ ′ 2 C B y C ′ B’’ ′′ A ′′ ′ -2y A B C’’ below thex-axis 2 4 ) C C C

′ 4 6 x x -1〉, , 123 6. 8. MP.2 MP.6 (

COMMON x, by afactor of 2. rotated 180°about origin, and the dilated then is translated△ABC along vector the 〈-1, C has thesameorientation. the sameshapeaspreimage butlarger. It doubles thesidelengths. The finalimageis changing theorientation. The dilation I andIVto Quadrants IIandIIIwithout the originmoves thefigure from Quadrants the shapeororientation. The rotation about figure withoutchanging down andto theleft Possible answer: - CORE ′ y - 8 )

Reasoning Precision 8 → - B C - 6 ′′′ (

′′′ Mathematical Practices - 6 A _ 2 3 - ′

x, - 4 B C _

3 2 4 ′′ ′′ C

y - ′′ )

C - 2 → ′′′ The translation moves the 2 ------8 6 4 2 2 4 6 A (

0 A x ′ 6 4 2 2 4 6 8 ′′′ B 0 + 6,y A A A y ′′′ ′′ A B y ′′ ′ 2 2 - 4 C B ′ ′ 4 )

C B A B → ′′′ -3〉, 6 (

Lesson 1 Lesson

_ 2 3 6 B

C x, x ′′ x - 5/14/14 5:01PM _ 2 3

y )

124 kx, ( ” this → ) undo x, y (

) y

k 1 __

k 1 __ ( → ) x, y ( Have students use the algebraic representation representation use the algebraic students Have Sequences of Transformations When writing the algebraic rules for the rigid rules for the algebraic writing When is the scale factor. Ask students how they how students Ask the scale k is , where factor. ) INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES MP.1 MP.4 Connections on Math Focus Focus on Modeling Focus motions and other transformations, review the transformations, other and motions should Students coordinates. and quadrants is angle a positive through a rotation that remember a rotation and direction, in the counterclockwise in the direction. clockwise is angle a negative through as plane in the coordinate a dilation of “ would that the dilation represent would dilation. ky DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 5/14/14 5:03 PM

No No No No

A sequence of a rotation and a dilation will a dilation and a rotation of A sequence image in an result always/sometimes/never the as orientation size and the same is that preimage. -axis the x-axis a reflection across of A sequence the y-axis then a reflection across and in a 180° results always/sometimes/never the preimage. of rotation A double rotation can always/sometimes/never always/sometimes/never can rotation A double rotation. a single as be written

Yes Yes Yes Yes 18. 16. 14.

124 of transformations. transformations. of

be used to create the image, the image, be used create to -4〉 -4〉. -4〉. ? Select yes or no. △LMN? Select or yes the preimage, , from under a sequence △LMN under of the image is each of the following sequences the following of each -axis and then Reflect and the y-axis across Rotate 180° about the reflect origin, 180° about Rotate translate along the vector 〈0, the vector along translate the y-axis. then reflectand across the about 90° clockwise Rotate and the x-axis, reflectorigin, across 90° counterclockwise then rotate the origin. about then translate and the x-axis, across 〈0, the vector along Translate along the vector 〈0, the vector along Translate

F E B A a. c. d. b. △QRS Can △QRS A sequence of rigid transformations will rigid transformations of A sequence is that image in an result always/sometimes/never the preimage. as orientation size and the same A sequence of a translation and a reflection and a translation of A sequence does not that a point has always/sometimes/never change position. A combination of two rigid transformations on a on rigid two transformations of A combination the produce will always/sometimes/never preimage order. when taken in a different image same reflected over a vertical line and a vertical and line G reflected the sequence: over of the result is line. then a horizontal around 90° clockwise D rotated the sequence: of the result is line. a horizontal then reflected vertices and over its of one 90° then rotated and translated the E sequence: of the result is counterclockwise. 90° counterclockwise rotated the D sequence: of the result is thenand translated.

19. Module 3 17. 15. Choose the correct word to complete a trueChoose statement. complete to the correct word 13. 9. 10. 11. 12. In Exercises 9–12, use Exercises the diagram.In Fill in the blank with the letter the correct described. image of

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y PEERTOPEER DISCUSSION 20. A teacher gave students this puzzle: “I had a triangle with vertex A at (1, 4) and vertex B at (3, 2). After two rigid transformations, I had the 6 Translation image shown. Describe and show a sequence of transformations that A′ A′′ Ask students to discuss with a partner how to A will give this image from the preimage.” 4 predict the final image for a combination of rigid B B Possible answer: Translate by the vector 〈2,1〉 then reflect Preimage ′ ′′ Reflection 2 B transformations and nonrigid transformations. Then over the line x = 5. C′ C′′ x ask students to make a conjecture about the result of 0 2C 4 6 dilating a triangle whose vertices have coordinates (1, 1) , ( 1, 4) , ( 5, 1) with a scale factor of 2, followed H.O.T. Focus on Higher Order Thinking by a reflection in the x-axis. The image has 21. Analyze Relationships What two transformations would y A vertices ( 2, 2) , ( 2, 8) , ( 10, 2) . – – – you apply to △ABC to get △DEF? How could you express these 4 transformations with a single mapping rule in the form of (x, y) → ( ?, ?) ? 2 C B x JOURNAL Possible answer: Reflect △ABC across the y-axis and then translate it down 7 units. A single mapping rule would be -4 -2 0 2 4 D Have students compare and contrast the methods (x, y) → (-x, y - 7) . -2

they have learned for combining rigid -4 transformations and nonrigid transformations in the F E coordinate plane. 22. Multi-Step Muralists will often make a scale drawing of an art piece before creating the large finished version. A muralist has sketched an art piece on a sheet of paper that is 3 feet by 4 feet. a. If the final mural will be 39 feet by 52 feet, what is the ©AJP/ scale factor for this dilation? Scale factor: 13

b. The owner of the wall has decided to only give permission to paint on the lower 1 half of the wall. Can the muralist simply use the transformation x, y x, __ y ( ) → 2 in addition to the scale factor to alter the sketch for use in the allowed space?( ) Explain. Only if the artist wants the final version of the mural to be distorted. This mapping will shrink the height of the mural in half, but by keeping the original width, the shapes will change. 23. Communicate Mathematical Ideas As a graded class activity, your teacher asks your class to reflect a triangle across the y-axis and then across the x-axis. Your classmate gets upset because he reversed the order of these reflections and thinks he © Houghton Mifflin Harcourt Publishing Company • Image Credits: Shutterstock will have to start over. What can you say to your classmate to help him? The order of these two reflections does not matter. The resulting image is the same for a reflection in the y-axis followed by a reflection in the x-axis as for a reflection in the x-axis followed by a reflection in the y-axis.

Module 3 125 Lesson 1

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125 Lesson 3 . 1 126 Sample answer: Each Sample answer: . Sequences of Transformations A visual pattern can be described as a form or be can described or form a A visual as pattern sequences show or say, write, to students Ask CRITICAL THINKING INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES half of any one of the arms can be taken as a pattern a pattern as of the arms can be taken one half of any times in the twelve approximately repeats that arm. on each twice design, snowflake Ask students whether a snowflake is a whether is a snowflake students Ask the them consider object. Have two-dimensional symmetry, of lines on dimension effects the third of view a side from appears the snowflake how and view. top a than rather MP.3 MP.4 on Communication Focus Focus on Modeling Focus shape that repeats. Ask students to describe to the students Ask repeats. that shape patterns. of in terms snowflake their using translations, and reflections, rotations, of example: For hands. thumbs Reflection outward, facing — both hands nearly together one and forward facing hand — one Translation orientation. in the same backward Rotation up) left (thumb pointing hand — one fingers down), (thumb right pointing hand one and other. each facing Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: 3/20/14 5:26 PM

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mark Cassino/Superstock Lesson Lesson 1 126 The photograph shows an actual snowflake. Draw a detailed sketch of the “arm” of of the “arm” of a detailed sketch Draw actual snowflake. an shows photograph The

EXTENSION ACTIVITY EXTENSION Have students research the claim that “all snowflakes are different.” Depending different.” are snowflakes “all that the claim research students Have where and how investigate lead them to may the claim interests, students’ upon they why they as the fall atmosphere, through they change how form, snowflakes have humidity and temperature the effects and that structure, a hexagonal have their structure. upon

Module Module 3 The entire flake can be created by rotating the arm through 60°, 120°, 180°, 60°, 120°, the arm through rotating by can be created flake entire The above mentioned “ear” the arms, of the new several 240°, and 300°. For of times as translations or it appears several form, dilated appears in a slightly one another. In their descriptions, students should refer to specific features of their features specific to refer should students In their descriptions, dividing the 10:00 arm in half is a line of reflection, line with The drawings. the portion on each side being (nearly) a reflection of the other of the flake in the “ear” a small imperfection in this description, with the large There’s side. it should be. image where a mirror having side not quite middle of the right reflecting the by on the other side can its almost-image be created However, downward. it slightly the line of symmetryear across and then translating the snowflake located at the top left of the photo (10:00 on a clock face). Describe a clock (10:00 on in as the photo left of the top at located the snowflake see. you that rotations or reflections, translations, any can you detail as much in the found you based what constructed, is on snowflake the entire describeThen how arm. one design of drawings. Check students' Lesson Lesson Task Performance

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