Sketching Transformations

I have come to recognize that many of my senior math students do not make a connection between transformations and algebraic processes.  They struggle to make the connection between physical movement and algebraic effect.  This activity is intended to assist these students in building connections in this area.

  1. Step 1 – Review Cartesian planes and coordinate systems
    1. Access prior knowledge
    2. Identify any misconceptions or gaps in learning that should be addressed in these areas before moving forward.
  2. Step 2 – Student Hypothesis
    1. Working in pairs, students will create rules for translations, reflections, and rotations while working with coordinates (e.g. when translating horizontally, the x-coordinate of each point is adjusted accordingly)
    2. While working on developing their hypotheses, students have access to a tabletop grid and shape cutouts to manipulate
  3. Step 3 – Test hypothesis
    1. Use Geometer’s Sketchpad to test hypothesis using prescribed “test” transformations
    2. Students access transformations from a list within the Geometer’s Sketchpad file
    3. Transformations will include confounding situations, such as a rotation in the opposite direction or a reflection in something other than an axis.
  4. Step 4 – Refine rules/hypothesis in consultation with a small group
    1. Students combine into groups of 3-5 students to discuss findings, inconsistencies, confirmations, etc
    2. Groups come to an agreed upon set of rules by discussing and justifying their perspectives.
  5. Step 5 – Use transformation rules to design a patterning activity for other groups in class
    1. Each group will use Geometer’s Sketchpad to design a problem scenario involving transformations that will require other groups to apply their transformation rules.
    2. Examples of problem scenarios could include designing a quilt pattern, wall or floor tiling, yard landscaping, etc. The scenario context will be an essential component of the framing of the application because “contexts allow the learner to reflect on and control for the meaning and reasonableness of their developing ideas” (Dixon, 1997, p. 140).

Dixon, J. K. (1997). Computer use and visualization in students’ construction of reflection and rotation concepts.School Science and Mathematics, 97(7), 352-358.


  1. Thank you Stephanie for a mathematics lesson with your references to Dixon’s work. The lesson begins from your observations of students as they struggle to make the connection between physical movement and algebraic effect. The example such as (e.g. when translating horizontally, the x-coordinate of each point is adjusted accordingly) helps to bring it to light what your class might construct in terms of hypotheses that later can be tested with Geometer’s Sketchpad. In Step 4, do you envision that a compendium of rules might be put forward, since each group might have tested a different hypothesis? The nice part of the concluding activity is that students, who have constructed or deconstructed their own rules then get to apply their rules to a new problem like making a quilt.

    Thanks for introducing us to Geometer’s Sketchpad! Samia

    1. Hi Samia,
      I envision that some of the rules will ultimately be common across the groups, but that the differences between groups will provide a form of distributed cognition, in which all students can benefit and learn from the work of all of their peers. For students who may not have the specific rules needed for the quilt, this scenario and the recognition that they need more information would hopefully encourage them to engage in discussion with peers and/or revisit their own work to fill in their identified gaps.

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