Getting Started 
Misconception/Error The student does not understand similarity in terms of transformations. 
Examples of Student Work at this Level The student:
 Attempts to determine whether or not the triangles are similar by investigating the proportionality of the sides and the measures of the angles.
 Describes a single transformation and, upon questioning, does not appear to understand similarity in terms of transformations.

Questions Eliciting Thinking What does it mean to show two triangles are similar using similarity transformations?
Can you explain why you performed this transformation? What were you trying to achieve? 
Instructional Implications If needed, assist the student in developing an understanding of dilations by using 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). Be sure the student understands the role of both the center and the scale factor in performing dilations. Provide opportunities for the student to apply the definition to dilate a variety of figures beginning with simple figures such as points, segments, and angles. Review the Fundamental Theorem of Similarity and the properties of dilations:
 Dilations map lines to lines, rays to rays, and segments to segments.
 The distance between points on the image is the scale factor times the distance between the corresponding points on the preimage.
 A dilation maps a line containing the center of dilation to itself and every line not containing the center of dilation to a parallel line.
 Dilations preserve angle measure.
Review the definition of similarity in terms of similarity transformations. Explain that two triangles are similar if there is a dilation or a dilation and a congruence (i.e., a sequence of rigid motions) which carries one triangle onto the other. Provide the student with two similar triangles that are related by dilation and have the student determine the center of dilation and the scale factor. Given two similar triangles that are related by a dilation followed by a sequence of rigid motions, have the student determine the scale factor and center of dilation and the rigid motions that will map one triangle onto the other. Provide assistance as needed. 
Moving Forward 
Misconception/Error The student incorrectly or incompletely describes similarity transformations. 
Examples of Student Work at this Level The student understands the need to describe a sequence of transformations that carries one triangle to the other in order to show the triangles are similar. However, the sequence the student describes is incorrect or incomplete.

Questions Eliciting Thinking What does it mean to show two triangles are similar using similarity transformations?
Can you show me how you would perform the transformations you described? Will these transformations carry one triangle to the other? 
Instructional Implications Ask the student to perform the transformations he or she described and provide feedback. Assist the student in revising the sequence so that one triangle is mapped to the other. Guide the student to describe each transformation in the sequence completely and clearly.
Provide additional pairs of similar triangles. Have the student justify the similarity of each pair of triangles by describing a sequence of similarity transformations. 
Almost There 
Misconception/Error The student does not completely and concisely describe the transformations that map one triangle onto the other. 
Examples of Student Work at this Level The student describes a sequence pf transformations that will carry one triangle to the other but omits a component of the description. For example, the student does not include the center or degree of rotation, does not include the center of dilation or its scale factor, or does not make clear which triangle is being mapped onto the other.

Questions Eliciting Thinking What should be included in your description of the rotation (reflection, or dilation)?
Which triangle was transformed? 
Instructional Implications Review what is necessary to completely describe each transformation, that is, the direction and magnitude of a translation; the line of reflection; the center, degree, and direction of rotation; and the center and scale factor of a dilation. Ask the student to revise his or her response to complete each description.
Provide additional pairs of similar triangles. Have the student justify the similarity of each pair of triangles by describing a sequence of similarity transformations. If possible, have students trade descriptions to see if each can be followed without further explanation. 
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 determines that and justifies the similarity by describing a sequence of similarity transformations that maps one triangle onto the other. For example, the student describes the following sequence:
 Rotate 180°clockwise (or counterclockwise) about point B, so that (3,1), (3,2), (1,2).
 Dilate about point B with a scale factor of two, so that (7,4), (7,2), (1,2).
The student understands that because a sequence of similarity transformations carries to , the triangles are similar. 
Questions Eliciting Thinking Could you perform the sequence of similarity transformations you described without using the coordinate plane?
Is it always necessary to include a dilation when describing the sequence of similarity transformations used to verify two triangles are similar? Why or why not?
How is the scale factor of the dilation related to the ratio of the corresponding sides?
Does it matter which transformation is done first? 
Instructional Implications Ask the student if it is possible to construct a triangle with angles that are congruent to which is not similar to . Have the student explore this possibility by constructing several triangles with varying side lengths that have angles that are congruent to those of . Ask the student to determine whether or not the triangles are similar. Then ask the student to identify a sequence of similarity transformations that could be used to justify the similarity.
Consider implementing MFAS task Showing Similarity (GSRT.1.2) and/or The Consequences of Similarity (GSRT.1.2). 