Transformations Lesson 3

  • Exploring Rotations

    Lesson 3 of 4
    8th grade

    75-90 minutes

    Description

    Explore rotations on a coordinate plane where (0,0) is the center; determine general rules for rotations.

    Materials

    Introduce

    Using the image below, have students “I see, I think, I wonder” about the rotation shown. Have a discussion using these phrases to start students thinking about what happens to a figure during a rotation.


    grid displaying two rectangles that are rotations of each other
    From Introduce/Engage handout (download from Materials section above)

    Teacher Notes:

    Possible questions to ask include:

    • What are some characteristics that you noticed about rotations?[Possible student responses: Moved two quadrants, shape size is preserved, it’s upside down]
    • What happens to the figure that you rotate? What changes? [Possible student responses: the orientation and location of the figure changes with rotations, but the figures are still congruent. ]
    • Do you notice any patterns? What is preserved? [Possible student responses: with each rotation the size and shape are preserved.]
    • What else do you notice about the rotations? [Possible student responses: answers may vary]

    Explore

    In pairs, students will explore and draw rotations using patty paper or tracing paper on their Rotations Activity Sheet (download from Materials section above). Students will trace and label the image and axis from the grid onto patty paper. With their pencil on origin, (0,0), as the center of rotation, they will rotate the figure the indicated direction and number of degrees, and label new points using appropriate prime notation.

    Partners will work together to complete the tables and notice differences and similarities between the pre-image and the rotated image by noting observations and patterns. Possible questions are provided.

    Teacher Notes:

    Possible questions to ask include:
    If you rotate a figure 90°, clockwise/counterclockwise, on the coordinate graph, what would happen to the coordinates?

    • clockwise: If the original point is (x,y), then the point after rotation will be (-x, y)
    • counterclockwise: If the original point is (x,y), then the point after rotation will be (-y, x)

    If you rotate a figure 180°, clockwise, on the coordinate graph, what would happen to the coordinates?

    • clockwise: If the original point is (x,y), then the point after rotation will be (-x, -y)
    • counterclockwise: If the original point is (x,y), then the point after rotation will be (-x, -y)

    If you rotate a figure 270°, clockwise/counterclockwise, on the coordinate graph, what would happen to the coordinates?

    • clockwise: If the original point is (x,y), then the point after rotation will be (-y, x).
    counterclockwise: If the original point is (x,y), then the point after rotation will be (y, -x)

    Synthesize

    As a whole group, have the class summarize the patterns they noticed and the generalizations for the changes to an image during a rotation. Share and chart observations, highlighting student answers that lead to general rules for rotation.

    Assessment

    Exit Ticket:
    Perform a 180° clockwise rotation around the origin, and describe the rotation using as much detail as possible.


    grid with two triangles that are rotations of each other
    Figure from optional Rotations Assessment sheet (download from Material section or links above)

    Teacher Reflection

    • How were you able to gather and elicit evidence of student thinking and understanding from exploring rotating clockwise and counterclockwise? How could that evidence help during the instruction for developing general rules for rotating figures 90°, 180°, and 270°, both counterclockwise and clockwise?
    • How did you connect student-generated strategies and methods to the more formal procedure of creating rules for rotations?
    • What scaffolding or differentiation was needed in response to student thinking? How did you extend learning for students?

    Leave your thoughts in the comments below.

    Other Lessons in This Activity

    Lesson 1 of 4

    Develop an understanding of translations in the coordinate plane and determine general rules for translations through exploration.

    Lesson 2 of 4

    Develop an understanding of reflections in the coordinate plane, and determine general rules for reflections through exploration.

    Lesson 4 of 4

    Students will develop an understanding of dilations on the coordinate plane and general rules for dilating an image through exploration.

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  • Essential Question(s)

    • What are general rules and patterns when rotating images 90°, 180°, 270° clockwise and counterclockwise?
    • What properties are preserved during rotation?

    Standards

    CCSS, Content Standards to specific grade/standard

    • 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:
      • Lines are taken to lines, and line segments to line segments of the same length.
      • Angles are taken to angles of the same measure.
      • Parallel lines are taken to parallel lines.
    • 8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

    CCSS, Standards for Mathematical Practices

    • SMP 5 Use appropriate tools strategically.
    • SMP 8 Look for and express regularity in repeated reasoning

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Build procedural fluency from conceptual understanding.
    • Elicit and use evidence of student thinking.