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Representing and Combining Transformations
During the
Lesson
  • Whole-Class Interactive Discussion (15 minutes)
      • Give each student the transparency L-Shapes, and a pin (to help find centers of rotation).

      Using transparencies encourages students to test different transformations. Working dynamically should deepen students' understanding of transformations in a way that simply drawing shapes on a graph does not.

      • Introduce the lesson by using slides P-1, P-2, and P-3 of the projector resource.
      • Ask the students where they think the image of the L-Shape will be after it has been translated, reflected, or rotated in different ways:
        • Where will the L-Shape be if it is translated −2 units horizontally and +1 units vertically?
        • Where will the L-Shape be if it is reflected over the line x = 2?
        • Where will the L-Shape be if it is rotated through 180° around the origin?
      • Ask volunteers to demonstrate their answers by placing their grid and L-Shape on the overhead projector.
      • Discuss these positions with the rest of the class, and encourage students to challenge their peers if they think the L-Shape has been positioned incorrectly.
      • Once the correct position has been agreed on, move on to the next transformation.
      • You may also want to move the L-Shape to a different position on the grid, and ask students:
        • What transformation will move the L-shape to this new position? Show me.
  • Collaborative Work (30 minutes)
      • Ask students to work in groups of two or three.
      • Give each group Card Set A: Shapes and Card Set B: Words and a copy of the transparency Transformations.
      • Introduce the activity:
        • You are now going to continue to transform L-shapes.
        • You've got six shape cards, each showing a different L-shape, and eight word cards, each of which describes a different transformation.
        • Take turns to link two shape cards with a word card. Make sure the arrow goes in the right direction! Each time you do this, explain your thinking clearly and carefully.
        • Your partner should then either explain their reasoning again in his or her own words or challenge the reasons you gave.
        • It is important that everyone in the group understands the placing of a word card between two shape cards.
        • Ultimately, you want to make as many links as possible. Use all the shape cards and all the word cards if possible.
      • You may wish to use Slide P-4 of the projector resource to display these instructions.
      • You have two tasks during the paired work: to make a note of student approaches to the task, and to support student reasoning.

      Make a note of student approaches to the task.

      • Listen and watch students carefully.
      • In particular, listen to see whether students are addressing the difficulties they experienced in the assessment.
      • For example, are students having difficulty rotating a shape around (2, 0) or reflecting a shape over the lines y = x and y = −x?
      • You can use this information about particular difficulties as a focus for a whole-class discussion later in the lesson.

      Support student reasoning.

      • Use the questions in the Common Issues table to help address misconceptions.
      • Encourage students to explain carefully how they have made each connection.
        • Lian, please explain why you've linked these two shapes with this transformation.
        • Laura, can you repeat Lian's explanation in your own words?

      Ask students:

        • How does folding the L-Shape along the line of reflection help when reflecting the shape?
        • How does drawing a line from the center of rotation to a corner of the shape help when rotating the shape?
      • Students who are struggling should be encouraged to concentrate on linking Shape Cards A, B, C, and D.

      Further Transformations

      • Once students have completed their arrangement of cards, give them a copy of Card Set C: Additional Words and a pair of scissors.
      • Ask students to add an appropriate transformation, where possible, between any shape cards that has not yet been connected.
      • On completion, students may then glue the cards on a poster. They will need a glue stick and a sheet of large poster paper to do this.

      Extension Task

      • If a group of students successfully completes the task:
        • Can you find a combination of two transformations that could be replaced by a single one?
        • [For example, reflect B over the x-axis B onto A then reflect A over the y-axis onto C. These two transformations can be replaced by a single transformation: rotate B through 180° around the origin onto C. This can be seen on the example arrangement below.]
      • Students should be encouraged to investigate whether or not this is always the case.
        • For any shape, will this combination of transformations always replace this single one?

      A proof would involve considering what would happen to the general point (x, y). Under a reflection over the x-axis, this would go to (x, −y). After a further reflection over the y-axis, this would become (−x, −y). This is the same as the general point (x, y) being rotated through 180° around the origin.

      • Students should be encouraged to look for other possible combinations in their card arrangements in the same way.
  • Whole-Class Discussion (15 minutes)
      • Give each group of students either a mini-whiteboard, pen, and eraser, or a piece of square paper.
      • Use Slides P-5 and P-6 of the projector resource to support a whole-class discussion.
      • Ask students to do the following transformations using the coordinate grid on the transparency Transformations, then to write the new coordinate on their mini-whiteboard.
        • Use the transparency, Transformations. Mark the coordinate (1, 4) on the coordinate grid.
        • Show me the new coordinates of the point (1, 4) after it is:
          • Reflected over the x-axis. (1, −4)
          • Reflected over the y-axis. (−1, 4)
          • Rotated through 180° around the origin. (−1, −4)
          • Reflected over the line y = x. (4, −1)
          • Reflected over the line y = -x. (−4, −1)
          • Rotated through 90° clockwise around the origin. (4, −1)
          • Rotated through 90° counterclockwise around the origin. (−4, 1)
      • You may like to repeat this with a general starting point (x, y).
        • Show me the new coordinates of the general point (x, y) after it is:
          • Reflected over the x-axis. (x, −y)
          • Reflected over the y-axis. (−x, y)
          • Rotated through 180° around the origin. (−x, −y)
          • Reflected over the line y = x. (y, x)
          • Reflected over the line y = −x. (−y, −x)
          • Rotated through 90° clockwise around the origin. (y, −x)
          • Rotated through 90° counterclockwise around the origin. (−y, x)
      • It may be helpful to write the new coordinates on the board to be able to extend discussions to include combinations of transformations.
        • What is the single transformation that will produce the same result as:
          • A rotation of 90° clockwise around the origin followed by a reflection in the y-axis?
          • [This is a reflection in the line y = −x.]
        • Show me two transformations that can be written as a single direction.
        • Show me two transformations that cannot be written as a single direction. Can you change the starting point of the shape so that it can be written as a single direction?
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