Getting Started 
Misconception/Error The student is unable to accurately describe a sequence of transformations that demonstrate similarity. 
Examples of Student Work at this Level The student:
 Describes only one of the transformations (e.g., dilation or reflection).
 Describes an incorrect transformation (e.g., translation).
 Writes the figures are similar because they have the same shape but a different size, with no reference to transformations.

Questions Eliciting Thinking What is a transformation? Can you think of any examples of transformations?
Can you define the word similar in terms of transformations?
How might you tell if two figures are similar? Can you explain this in terms of transformations? 
Instructional Implications If needed, review the rigid motions (translations, reflections, and rotations) and the definition of congruence in terms of rigid motion. Consider implementing the MFAS tasks to assess the studentâ€™s understanding of congruence.
Review the definition of dilation. Have the student use graph paper and a ruler, dynamic geometry software, or interactive websites (e.g., ) to obtain images of figures under dilations having specified centers and scale factors. Allow the student to explore dilations using handson activities.
Review the definition of similarity in terms of similarity transformations. Explain that two figures are similar if there is a dilation or a dilation and a sequence of rigid motions that will carry one figure onto the other. Assist the student in applying the definition of similarity in terms of similarity transformations to show that two figures are similar. Provide the student with two similar figures (e.g., a pair of triangles or a pair of quadrilaterals) that are related by a dilation and have the student determine the center of the dilation and the scale factor. Explain to the student that describing the dilation in detail (e.g., by specifying its center and scale factor) and then performing the dilation is a convincing way to show the two figures are similar. Next, provide two similar figures that are related by a dilation and a rigid motion. Have the student describe both the dilation and the rigid motion that will carry one figure onto the other. Ask the student to perform the transformations to ensure that the figures are similar. Provide assistance as needed. 
Moving Forward 
Misconception/Error The student provides a general description of the sequence of transformations that demonstrate similarity. 
Examples of Student Work at this Level The student:
 Simply writes, â€śreflection then dilationâ€ť or â€śdilation then reflection.â€ť
 Makes no reference to the proportional change in size.
The student does not make clear that the two figures coincide.

Questions Eliciting Thinking Can you describe the reflection more specifically? What is the line of reflection?
Can you describe the dilation more specifically? What is the scale factor? Where is the center of dilation?
How do these two transformations show similarity? 
Instructional Implications With regard to reflections, be sure the student understands that a reflection is a transformation of the plane. A reflection across line m (the line of reflection) assigns to each point not on line m, a point that is symmetric to itself with respect to line m. Discuss the basic properties of reflections (e.g., reflections map lines to lines, rays to rays, and segments to segments; reflections are both distance preserving and degree preserving) and how these properties ensure that the image of a figure under a reflection is always congruent to the preimage.
With regard to dilations, be sure the student understands that a dilation is a transformation that enlarges or reduces an image by a scale factor about a fixed point (called the center of dilation). Discuss the basic properties of dilations (e.g., angle preservation, parallelism, colinearity, and orientation) and how these properties ensure that the image of a figure under a dilation is always similar to the preimage.
Encourage the student to be precise when describing transformations. Model a concise description using mathematical terminology. Then make clear that the figures are similar because the sequence of transformations carries one figure onto the other.
Provide additional opportunities to show that two figures are similar by describing specific transformations that carry one figure onto the other. Remind the student to include all necessary components in each description, identifying the center and scale factor of the dilation, the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. 
Almost There 
Misconception/Error The studentâ€™s description contains a minor error. 
Examples of Student Work at this Level The student provides a detailed description of the sequence of transformations that demonstrate similarity, but the description contains an error. For example, the student:
 Describes the scale factor incorrectly.
 Describes the center of dilation incorrectly.
 Describes the line of reflection incorrectly.

Questions Eliciting Thinking Are you comparing the dimensions of the trapezoid or the area of the trapezoid? Does it matter? To what does scale factor refer?
How can you specify the line of reflection? 
Instructional Implications Provide specific feedback to the student concerning any error made and allow the student to revise his or her work. Encourage the student to check his or her work for accuracy.
Provide additional opportunities to show that two figures are similar by describing specific transformations that carry one figure onto the other. Remind the student to include all necessary components in each description, identifying the center and scale factor of the dilation, the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student describes a reflection of trapezoid ABCD across the xaxis, and a dilation with a scale factor of two and a center at point D. The student explains that since the figures now coincide, they are similar. 
Questions Eliciting Thinking How does describing this sequence of transformations ensure that the figures are similar?
If two figures are similar, what must be true of their corresponding angles? What property of rigid motions ensures this?
If two figures are similar, what must be true of corresponding sides? What aspect of the sequence of transformations ensures this?
Are congruent figures necessarily similar? Are similar figures necessarily congruent? 
Instructional Implications Consider implementing the MFAS task Similarity  3. 