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Sequences of Transformations 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 LESSON3 . 1 Name Class Date Sequences of 3 . 1 Sequences of Transformations Transformations Essential Question: What happens when you apply more than one transformation to a figure? Resource Common Core Math Standards Locker The student is expected to: Explore Combining Rotations or Reflections COMMON A transformation is a function that takes points on the plane and maps them to other points on CORE G-CO.A.5 the plane. Transformations can be applied one after the other in a sequence where you use the ... Specify a sequence of transformations that will carry a given figure image of the first transformation as the preimage for the next transformation. onto another. Also G-CO.A.2, G-CO.B.6 Find the image for each sequence of transformations. Mathematical Practices Using geometry software, draw a triangle and label the COMMON vertices A, B, and C. Then draw a point outside the A CORE MP.5 Using Tools ″ B A′ triangle and label it P. ″ B′ A Language Objective Rotate △ABC 30° around point P and label the image as A B C Then rotate A B C 45° around point P and C″ Explain to a partner why a transformation or sequence of transformations △ ′ ′ ′. △ ′ ′ ′ C B label the image as A B C . Sketch your result. ′ is rigid or nonrigid. △ ″ ″ ″ C P ENGAGE Make a conjecture regarding a single rotation that will map △ABC to △A″B″C″. Check your conjecture, and describe what you did. What happens Essential Question: A rotation of 75° ( because 30 + 45 = 75) should map △ABC to △A″B″C″. By using the when you apply more than one software to rotate △ABC 75°, I can see that this image coincides with △A″B″C″. transformation to a figure? Using geometry software, draw a triangle and label the Possible answer: The transformations occur j vertices D, E, and F. Then draw two intersecting lines and D E E D k sequentially, and order matters. The result may be label them j and k. ′ ′ D″ the same as a single transformation. Reflect △DEF across line j and label the image as F E″ △D′E′F ′. Then reflect △D′E′F ′ across line k and label the F′ image as △D″E″F ″. Sketch your result. F″ PREVIEW: LESSON PERFORMANCE TASK © Houghton Mifflin Houghton © Company Harcourt Publishing View the Engage section online. Discuss the photo Consider the relationship between △DEF and △D″E″F″. Describe the single transformation that maps DEF to D E F . How can you check that you are correct? and ask students to describe the snowflake in general △ △ ″ ″ ″ A rotation with center at the intersection of j and k maps △DEF to △D″E″F″. Rotating △DEF terms, such as “It has six arms that look alike.” Then around the intersection of j and k by the angle made between the lines rotates it about preview the Lesson Performance Task. halfway to △D″E″F″, so rotate it by twice that angle to see △DEF mapped to △D″E″F″. Module 3 through "File info" 115 Lesson 1 DO NOT EDIT--Changes must be made CorrectionKey=NL-B;CA-B Date Class Name Sequences of Transformations transformation to a figure? 3 . 1 Resource Locker What happens when you apply more than one Essential Question: that will carry a given figure onto another. Also HARDCOVER PAGES 103112 COMMON CORE G-CO.A.5 … Specify a sequence of transformations G-CO.A.2, G-CO.B.6 Combining Rotations or Reflections Explore on the plane and maps them to other points on he other in a sequence where you use the A transformation is a function that takes points e for the next transformation. the plane. Transformations can be applied one after t GE_MNLESE385795_U1M03L1.indd 115 image of the first transformation as the preimagations. 6/19/14 12:54 AM A Find the image for each sequence of transform A B ′ ″ ″ B draw a triangle and label the ′ A nt outside the Using geometry software, B, and C. Then draw a poi A, C B Turn to these pages to vertices ″ C′ triangle and label it P. and label the imageand as C ABC 30° around point 45°P around point P Rotate △ ′ P Then rotate A B C. Sketch your result. △A′B′C ′. △ ′ ″′ label the image as △A″B″C find this lesson in the to △A″B″C″. single rotation that will map ABC △ B C″. By using the be what you did. ″ ″ Make a conjecture regarding a should map △ABC to △A hardcover student ) Check your conjecture, and descri ides with △A″B″C″. (because 30 + 45 = 75 A rotation of 75° , I can see that this image coinc k software to rotate ABC 75 j D △ ° E ′ D E ′ D″ draw a triangle andecting label lines the and E edition. Using geometry software, ″ D, E, and F. Then draw two inters vertices F F′ F label them j and k. image as ″ across line j and label the DEF ′ across line k and label the Reflect △ D E F ′. Then reflect △D′E′F △ ′ ′ ″. Sketch your result. image as △D″E″F single and D E F . Describe thethat you are correct? tween DEF △ ″ ″ ″ F . Rotating DEF △ ″E″ ″ △ © Houghton Mifflin Harcourt Publishing Company Publishing Harcourt Mifflin Houghton © to △D″E″F″. How can you check Consider the relationship bes △DEF and k maps △DEF to △D tersection of j the lines rotates it about transformation that map F . D″E″ ″ A rotation with center at the in k by the angle made between Lesson 1 le to see △DEF mapped to △ around the intersection of j and F so rotate it by twice that ang ″E″ ″, 115 halfway to △D 6/19/14 12:54 AM Module 3 03L1.indd 115 GE_MNLESE385795_U1M 115 Lesson 3 . 1 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" DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A Reflect 1. Repeat Step A using other angle measures. Make a conjecture about what single transformation EXPLORE will describe a sequence of two rotations about the same center. If a figure is rotated and then the image is rotated about the same center, a single rotation Combining Reflections by the sum of the angles of rotation will have the same result. 2. Make a conjecture about what single transformation will describe a sequence of three rotations INTEGRATE TECHNOLOGY about the same center. A sequence of three rotations about the same center can be described by a single rotation Students have the option of completing the by the sum of the angles of rotation. combining reflections activity either in the book or online. 3. Discussion Repeat Step C, but make lines j and k parallel instead of intersecting. Make a conjecture about what single transformation will now map △DEF to △D″E″F ″. Check your conjecture and describe what you did. △D″E″F″ looks like a translation of △DEF. I marked a vector from D to D″ and QUESTIONING STRATEGIES translated DEF by it . The image coincides with D E F , so two reflections in || lines △ △ ″ ″ ″ How can you use geometry software to check result in a translation. your transformations? For reflections in parallel lines, use the measuring features to see if all Explain 1 Combining Rigid Transformations points move the same distance in the same In the Explore, you saw that sometimes you can use a single transformation to describe the direction. For reflections in intersecting lines, rotate result of applying a sequence of two transformations. Now you will apply sequences of rigid transformations that cannot be described by a single transformation. the preimage figure to see if the images are the same size and shape. Example 1 Draw the image of △ABC after the given combination of transformations. Reflection over line ℓ then translation along ⇀v ⇀ν EXPLAIN 1 ℓ Mifflin Houghton © Company Harcourt Publishing A Combining Rigid Transformations C B AVOID COMMON ERRORS Some students may transform the original figure Step 1 Draw the image of ABC after a reflection Step 2 Translate A B C along ⇀v . △ △ ′ ′ ′ twice instead of transforming the first image to get across line ℓ. Label the image △A′B′C ′. Label this image △A″B″C ″. the second, and the second to get the third. Note that ⇀ ⇀ ν ν when performing two transformations with A A' C″ → C′ C ′ B″ as the first transformation, A is the preimage and A' is B ℓ B ′ ′ ℓ the image. In the second transformation A' → A", A' A′ A′ A″ " A A is the preimage and A is the image. C C B B Module 3 116 Lesson 1 PROFESSIONAL DEVELOPMENT GE_MNLESE385795_U1M03L1.inddMath Background 116 3/20/14 5:27 PM Students have worked with individual transformations and should now be able to identify and describe translations, reflections, and rotations. In this lesson, they combine two or more of these transformations and may include sequences of nonrigid transformations. They must be able to visualize and predict the outcome of performing more than one transformation, as well as consider other transformations that produce the same final image. Throughout the lesson they must recall the properties of each transformation and the methods for drawing them. Sequences of Transformations 116 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 ⇀ B 180° rotation around point P, then translation along v , then reflection B′′ QUESTIONING STRATEGIES across line ℓ B′ A′′ A After a rigid motion, an image has the same Apply the rotation.
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