Getting Started 
Misconception/Error The student does not understand similarity in terms of transformations. 
Examples of Student Work at this Level The student:
 Writes the corresponding angles are congruent and the corresponding sides are proportional because the triangles are similar.
 Explains in terms of the converse of the statement (i.e., if corresponding angles are congruent, then the triangles are similar).
 Writes that without a diagram, he or she cannot determine whether or not the corresponding angles are congruent and/or the corresponding sides are proportional.
 Writes that both the corresponding angles and the corresponding sides would be proportional.
 Indicates that he or she does not understand similarity in terms of transformations.

Questions Eliciting Thinking What does it mean for two triangles to be similar?
How is similarity determined or verified?
How can transformations be used to define and justify similarity?
How can you determine, based on a sequence of similarity transformations, that ?
Do similarity transformations preserve angle measure?
In a dilation, are the corresponding sides proportional?
What is scale factor, and how does it relate to similarity? 
Instructional Implications 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. Continue to explain that a dilation changes the lengths of the sides by a given scale factor, r, such that and therefore, ensures proportionality of corresponding sides. A dilation also preserves angle measure and therefore, ensures corresponding angles are congruent. Remind the student that rigid motions preserve lines, rays and segments and are both angle and distancepreserving. If the student is still struggling with his or her understanding of similarity in terms of similarity transformations, provide the student with several examples of similar triangles. Have the student identify the composition of transformations that maps one triangle onto the other. Lead the student to understand that because this composition of similarity transformations exists, the triangles are similar. Furthermore, because the triangles are similar, we know that there must be a composition of transformations that maps one triangle onto the other. Because similarity transformations preserve angle measures, we know that corresponding angles are congruent, and because dilations change side length by a given scale factor, we know that corresponding sides are proportional.
Consider implementing one of the following MFAS tasks Showing Similarity (GSRT.1.2), To Be or Not To Be Similar (GSRT.1.2).
If the student does not have a clear understanding of dilations, have the student develop his or her 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 ) to obtain images of a given figure under dilations having specified centers and scale factors. Have the student observe the changes in dilations with the same scale factor with centers that lie inside, on, and outside of the preimage. Review with the student the Fundamental Theorem of Similarity, which ensures that the sides of the triangle not lying on the center of dilation are parallel to the corresponding sides. The corresponding angles are therefore congruent because corresponding angles of parallel lines are congruent. 
Making Progress 
Misconception/Error The student understands the definition of similarity in terms of similarity transformations but omits important details or does not completely describe how this definition ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 
Examples of Student Work at this Level The student attempts to reason from similarity transformations to the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. However, the student does not directly or explicitly refer to similarity transformations (i.e., rigid motion and dilations) and their properties to explain why angle measure is preserved and lengths of sides are proportional.

Questions Eliciting Thinking What is the definition of similarity in terms of similarity transformations?
What properties are preserved in translations, rotations, and reflections?
What properties are preserved in dilations?
Can you describe in more detail how you know the corresponding angle measures are equal?
Can you describe in more detail how you know the corresponding side lengths are proportional? 
Instructional Implications Model for the student a clear and complete explanation of why the definition of similarity in terms of similarity transformations ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Explain that if , there is a composition of rigid motion and dilation that maps to . Since both rigid motion and dilation preserve angle measure, the angles of must be congruent to the corresponding angles of . Also, a dilation changes the lengths of the sides by a given scale factor, r, such that , and therefore, which ensures proportionality of corresponding sides.
Consider implementing MFAS task Congruence Implies Congruent Corresponding Parts (GCO.2.7) to reinforce the student’s understanding of rigid motions ensuring congruent corresponding parts. 
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 writes:
 Since , there is a composition of rigid motion and dilation that maps to . Since both rigid motion and dilation preserve angle measure, the angles of must be congruent to the corresponding angles of .
 Since dilations change lengths by a scale factor given by the dilation, there is some scale factor, r, such that AB = r(DE), BC = r(EF), and AC = r(DF). Consequently, AB:DE as BC:EF as AC:DF so that corresponding sides are proportional.

Questions Eliciting Thinking How are the definitions of congruence and similarity (in terms of transformations) alike, and how are they different?
Is there a sequence of dilations, or dilations and rigid motions that would produce an image that is not similar to the preimage? Why or why not? 
Instructional Implications Review/explain the Triangle Midsegment Theorem to the student. Ask the student to verify, using his or her understanding of a dilation, why the midsegment is parallel to and half the length of the third side of the triangle.
Consider implementing MFAS task Dilation of a Line Segment (GSRT.1.1). 