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Representing and Combining Transformations
Before the
  • Assessment Task: Transformations (15 minutes)
    • Set this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work, to find out the kinds of difficulties students have with it. You will then be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of the assessment task Transformations.
        • Read through the questions, and try to answer them as carefully as you can.

      It is important that, as far as possible, students are allowed to answer the questions without your assistance.

      Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to be able to answer questions like these confidently. This is their goal.

  • Assessing Students' Responses
    • Collect students' responses to the task, and make some notes on what their work reveals about their current levels of understanding. The purpose of doing this is to forewarn you of the difficulties students will experience during the lesson itself, so that you may prepare carefully.

      We suggest that you do not score students' work. The research shows that this will be counterproductive as it will encourage students to compare their scores, and will distract their attention from the mathematics.

      Instead, you can help students to make progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on the next page. These have been drawn from common difficulties in trials of this unit.

      We suggest that you write your own list of questions, based on your own students' work, using the ideas that follow. You may choose to write questions on each student's work. If you do not have time to do this, you could write a few questions that will help the majority of students. These can then be displayed on the board at the end of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student confuses the terms "horizontally" and "vertically."

      • For example: The student translates the shaded triangle −7 units vertically and +1 units horizontally in Q1a.
      • Look at the start of the word "horizontally." What are we referring to when we talk about the horizon? Which way is this?

      Student translates rather than reflect the shape (Q1b).

      • For example: The student has translated the shaded triangle vertically −7 units and so omitted to draw the mirror image.
      • If you were to place a mirror on the x-axis, what would the reflected image look like?

      Student confuses the terms "clockwise" and "counterclockwise."

      • For example: The student rotates the shaded triangle counterclockwise (Q1c.)
      • Think about the direction of the hands on a clock. This direction is "clockwise."

      Student ignores the center of rotation and rotates from a corner of the shaded triangle.

      • For example: The student rotates the shaded triangle around the point (1, 2) (Q1d.)
      • Where is the center of rotation?
      • Mark the center of rotation and draw a line to a corner of the shape. Where will this line be once it has been rotated?

      Student uses an inefficient combination of transformations.

      • For example: The student describes the transformation in Q2a as "a reflection over the y-axis, followed by a rotation 90° counterclockwise around (−1, 2), followed by a translation −1 unit horizontally and −3 unit vertically."
      • Is there a single transformation that will take the shaded triangle directly to the triangle labeled E?

      Student correctly answers all the questions.

      • The student needs an extension task.
      • Find a combination of two transformations that could be replaced by a single one.
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