Transformations Lesson 1

  • Exploring Translations

    Lesson 1 of 4
    8th grade

    40-60 minutes


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



    Prepare students to use the visual thinking strategy by providing the “I see, I think, I wonder” map on the board. Ask them to observe your actions and note their thinking. Demonstrate translation by picking up an object and physically moving it to a new location. Repeat if needed. Students should notice that each translation’s orientation, size, and shape are preserved. The only change with translations is location.

    Teacher Notes:

    Possible guiding questions include:

    • What are some characteristics that you noticed about the movements?

    [Possible student responses: the shape stayed the same, the object is not in the same spot, the object changed direction]

    • What changes do you notice?

    [Possible student responses: nothing about the object is different, it is just in a new place]

    • Are there any patterns that you notice?

    [Possible student responses: the shape always moves location but does not change in size]

    • What happens to the coordinates?

    [Possible student responses: they always change ]


    Have students work with their partner on Part I of the Translations Activity Sheet (download from Materials section above) to generalize a rule for each translation using x and y notation.

    After a few minutes, bring students back together to share student generalizations. Expand on student thinking to connect to the general translation rules using x and y notation. For example, (x +2, y-3) for a translation of a point 2 units to the right and 3 units down.

    Have students continue to work in partners to complete Part II Exploring Properties of Translations. For this part of the activity, students should be able to understand and apply a translation rule to a given image. 

    As a whole class, have partners share their observations.

    Teacher Notes:

    Possible guiding questions:

    • What do you notice about the pre-image and new image?

    [Possible student responses: The combination of these translations moved the image diagonally.]

    • How are all the points on the translated image related to the corresponding points on the pre-image?

    [Possible student responses: They are the same distance apart, just moved down.]

    • How do you know?

    [Possible student responses: All of the points on the shape are still the same distance away from each other.]


    After students have completed the second part of the worksheet, work as a class to summarize rules for performing a translation. Teachers may want to post the rules for translation on a classroom chart for future reference.

     Students should recognize the orientation, size, and shape are preserved in translations; that is, the resulting figure will have the same side lengths and interior angle measures as the original figure.

    Teacher Notes:

    Examples of rules include:

    • When translating up, the y-coordinate will increase by the number of units moved.
      • For example, if a point (x, y) is translated up 8 units, then the image point will be (x, y + 8).
    • When translating down, the y-coordinate will decrease by the number of units moved.
      • For example, if a point (x, y) is translated down 3 units, then the image point will be (x, y - 3).
    • When translating right, the x-coordinate will increase by the number of units moved.
      • For example, if a point (x, y) is translated right 4 units, then the image point will be (x+4, y).
    • When translating left, the x-coordinate will decrease by the number of units moved.
      • For example, if a point (x, y) is translated left 5 units, then the image point will be (x-5, y).


    Teacher Notes:

    Extension Questions

    • If you translate a figure three units to the right on the coordinate graph, how would the coordinates change? [three is added to each x-coordinate.]
    • If you to translate a figure 2 units down on the coordinate graph, how would the coordinates change? [2 is subtracted from each y-coordinate.]
    • How can you determine the coordinates of the points of the new figure given the translation without drawing the figure? [If a figure is translated left or right, a number should be added or subtracted from the x-coordinate, and if a figure is translated up or down, a number should be added or subtracted from the y-coordinate.]

    Teacher Reflection

    • How were you able to gather and elicit evidence of student thinking and understanding from “Part I Exploring Translations with Patty Paper”? How could this evidence help during instruction for developing a general rule for translations?
    • 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 2 of 4

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

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  • Essential Question(s)

    • What are general rules and patterns for translating an image?
    • What properties are preserved during translations?


    CCSS, Content Standards to specific grade/standard

    • 8.G.A.1Verify 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 1 Make sense of problems and persevere in solving them.
    • 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.