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 only or rotation only).
 Describes an incorrect transformation (e.g., reflection or translation).
 Writes that 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 MFAS tasks aligned to standard 8.G.1.2 to assess the student’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., http://www.mathsisfun.com/geometry/resizing.html, http://www.cpm.org/flash/technology/triangleSimilarity.swf ) 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.
Consider implementing the MFAS tasks for standard 8.G.1.1 and other MFAS tasks for standard 8.G.1.4. 
Moving Forward 
Misconception/Error The student provides only a general description of the sequence of transformations that demonstrate similarity. 
Examples of Student Work at this Level The student simply writes, “rotation then dilation” or “dilation then rotation.”
The student does not make clear that the two figures coincide.

Questions Eliciting Thinking Can you describe the rotation more specifically? Around which point are you rotating? What is the degree of the rotation? Clockwise or counterclockwise?
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 rotations, be sure the student understands that a rotation is a transformation of the plane. Each point in the plane is rotated a specified number of degrees (given by the degree of rotation) either clockwise or counterclockwise (indicated by the sign of the degree of rotation) about a fixed point called the center of rotation. Discuss the basic properties of rotations (e.g., rotations map lines to lines, rays to rays, and segments to segments; rotations are distance preserving; and rotations are degree preserving) and how these properties ensure that the image of a figure under a rotation 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 preserving, 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 student’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:
 Confuses the term clockwise with counterclockwise.
 Describes the degree of the rotation incorrectly (e.g., as 180° instead of 90°).
 Describes the scale factor of the dilation incorrectly (e.g., describes it as instead of or 2 instead of ).

Questions Eliciting Thinking What does clockwise mean? Which way do the hands of a clock move?
Can you demonstrate the rotation? How many degrees was that?
Are you comparing the dimensions of the parallelogram or the area of the parallelogram? Does it matter? To what does scale factor refer? 
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 90° counterclockwise rotation of quadrilateral ABCD about point C, and then describes a dilation of the image of quadrilateral ABCD with a scale factor of , and a center at point C. 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 tasks Similarity  2 and Similarity  3 (8.G.1.4). 