Transformations Lesson 2

  • Exploring Reflections

    Lesson 2 of 4
    8th grade

    60–90 minutes

    Description

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

    Materials

    Introduce

    Using the image below, have students “I see, I think, I wonder” about the reflection below. Have a discussion to start students thinking about what happens during a reflection.


    grid with 2 triangles that are a relfection of each other on the x=y line
    Image from Introduce/Engage Handout (downloadable from Materials section)

    Ask students to compare the distance and direction of each figure to the line of reflection. They should notice the distance from the line of reflection for each figure is the same, but in opposite directions. Introduce the formal vocabulary to describe this transformation as a reflection on the coordinate plane. Students should notice that in each reflection the size and shape are preserved. The orientation and location will change with a reflected image. They may also notice that the figures are congruent

    Teacher Notes:

    Lesson Plan ["Must Have"] Components

    There are multiple options to develop student understanding in reflections, including the use of Miras and patty paper.

    If using a Mira, place the Mira, beveled edge facing you and pointed down, on the line of reflection, and trace what you see in the Mira on the other side of the line of reflection.

    To use patty paper, trace the pre-image and line of reflection on the patty paper. Fold the paper on the newly traced line of reflection and trace the pre-image on the other side. A reflection image will be created.

    Possible questions to ask include:

    1. What are some characteristics that you noticed about reflections?” [Possible student responses: the shape stayed the same, the object is not in the same spot]
    2. What happens to the figure that you reflected? [Possible student responses: one point is the same while the other point is opposite ]
    3. What is preserved? What changes? [Possible student responses: length of line and angle are the same but their location is opposite the starting point]

    Explore

    Have students look at their Activity Sheet Part I. In pairs or small groups, they will explore and draw reflections using a manipulative (e.g. Mira, patty paper, etc.). As the pairs complete their drawings, ask them to discuss and provide observations in the space provided. As a whole class, share and chart observations. Highlight student observations that lead to the creation of rules for reflection.

    Teacher Notes:

    Example questions to prompt creation of rules:

    What are rules to describe the changes in coordinates when the image is reflected over the x-axis? The y-axis? The line y = x?

    1. When reflecting over the x-axis, the x-coordinate stays the same but the y-coordinate is the opposite of the original y-coordinate.
    2. For a point (x, y), after a reflection over the x-axis the new coordinates would be (x, -y).
    3. When reflecting over the y-axis, the x-coordinate is the opposite of the original x‑coordinate but the y-coordinate stays the same.
    4. For the point (x, y), after a reflection over the y-axis, the new coordinates would be (-x, y).

    What is the rule for a line of x = a, where a is any number on the plane? A line of y = b where b is any number on the plane?

    1. When reflecting over the y = x line:
      1. For the point (x, y), after a reflection over the y = x line, the new coordinates would be (y, x).

    In Part II Students will work in groups or pairs to explore what happens when they reflect a shape over a line of reflection of their choosing. The students will identify a line of reflection and record the coordinates before and after a reflection. Using the table, students should begin to identify patterns.

    Synthesize

    After students have completed the second part of the worksheet, work as a class to determine a set of rules for performing a reflection.

    Note that (1) some of the rules described are specific to certain lines of reflection and (2) that reflections preserve size and shape; that is, the resulting figure will have the same side lengths and interior angle measures as the original figure.

    Assessment

    Teacher Reflection

    • How were you able to gather and elicit evidence of student thinking and understanding from Part I Exploring reflections? How could this evidence help during the instruction for developing a general rule for reflections?
    • How did you connect the student-generated strategies and methods to the more formal procedure?
    • 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 3 of 4

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

    Lesson 4 of 4

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

  • Comments

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      Please note: the answers for Reflections Part I #3 do not match the student activity sheet.

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  • Essential Question(s)

    • What is a general rule for applying a reflection to an image?
    • Which properties are preserved during reflections?

    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.