Getting Started 
Misconception/Error The student does not understand similarity in terms of transformations. 
Examples of Student Work at this Level The student:
 Writes that the quadrilaterals are similar because their side lengths are proportional.
 Describes the dilation as being negative because it is a reduction.
 Identifies the scale factor but is unable to continue.
 Does not determine whether or not the figures are similar and describes only an incorrect sequence of transformations.
 Attempts to describe a sequence of similarity transformations that carries one quadrilateral onto the other but does not understand how this shows the two quadrilaterals are similar.

Questions Eliciting Thinking What does it mean for two quadrilaterals to be similar?
How can transformations be used to define and justify similarity? Which transformations can you use when showing two figures are similar?
What are the corresponding vertices of quadrilateral ABCD and quadrilateral EFGH? Are the corresponding angles congruent?
What do we know about corresponding sides of similar figures? Are the corresponding sides proportional? What is the ratio? How does the ratio relate to the scale factor? 
Instructional Implications Review the definition of similarity in terms of similarity transformations. Explain that two polygons are similar if there is a dilation or a dilation and a congruence (i.e., a sequence of rigid motions) which carries one polygon onto the other. 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. If needed, review with the student that the image resulting from the dilation of a segment passing through the center will lie on the same line as the original segment, while the dilation of a line segment that does not pass through the center will be parallel to the original segment (Fundamental Theorem of Similarity). Allow adequate time and handson activities for students to explore dilations visually and physically.
Discuss with the student how similarity transformations are used to define and determine similarity of figures. Explain to the student that because there is a dilation and a congruence that carries one quadrilateral onto the other, the quadrilaterals are similar, and because they are similar, the quadrilaterals have congruent corresponding angles and proportional corresponding sides. If needed, have the student measure the corresponding angles and sides of the original figure and its image to verify that the corresponding angles are congruent and the corresponding sides are proportional. Assist the student, if needed, in identifying the scale factor of the dilation. Show the student that each ratio of corresponding side lengths is equal to this scale factor. Present the student with several pairs of similar and nonsimilar polygons. Have the student determine whether or not each pair of polygons is similar describing the sequence of similarity transformations to justify his or her answer.
Provide the student with two similar figures (e.g., a pair of triangles or a pair of quadrilaterals) that are related by dilation and have the student determine the center of dilation and scale factor. Given two similar figures 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 rigid motions that will map one figure onto the other. Provide assistance as needed. 
Moving Forward 
Misconception/Error The student shows some understanding of the sequence of similarity transformations that maps one quadrilateral onto the other but omits important details or incorrectly describes the transformations in her or her explanation. 
Examples of Student Work at this Level The student:
 Identifies the types of transformations but does not provide enough detail. For example, the student says, “translate and then dilate the quadrilateral.”
 Incorrectly describes the translation and omits the center of the dilation.

Questions Eliciting Thinking Which transformations produce congruent figures? Which transformations produce similar figures?
Can you describe a sequence of similarity transformations that maps quadrilateral ABCD onto quadrilateral EFGH?
How would you describe a dilation? How is the center of dilation determined? What point did you use for the center of dilation?
How did you determine the scale factor? How did you determine the coordinates of the vertices of the image after the dilation?
How would you describe a translation? How did you determine the direction and distance of the translation? How did you determine the coordinates of the vertices of the image after the translation? 
Instructional Implications Review with the student the transformation(s) with which he or she struggled. With regard to dilations, be sure the student understands that a dilation is a transformation that moves each point of the preimage along a ray starting from a fixed center, and multiplies the distance from the center by a common scale factor. Further explain that when a side passes through the center of dilation, the side and its image lie on the same line and when a side does not pass through the center, the side and its image are parallel.
Guide the student through dilating the quadrilateral about the origin and again about one of the vertices of the quadrilateral. Ask the student to describe the similarities and differences between the two new images.
Present the student with several pairs of similar and nonsimilar polygons. Have the student determine whether or not each pair of polygons is similar describing the sequence of similarity transformations to justify his or her answer. 
Almost There 
Misconception/Error The student’s response is incomplete or contains unnecessary information. 
Examples of Student Work at this Level The student:
 Does not completely describe each transformation.
 Does not clearly describe which quadrilateral is being mapped onto the other.
 Does not specify the center of the dilation.

Questions Eliciting Thinking Can you describe in more detail the dilation? Where is the center of dilation? About what point did you dilate quadrilateral ABCD?
Can you describe the translation in more detail? 
Instructional Implications Review with the student the different similarity transformations and the terms most commonly associated with each (e.g., translationslide, rotationturn, reflectionflip, dilationreduction/enlargement). Encourage the student to use mathematical terms to describe specific transformations. Guide the student through dilating the quadrilateral about the origin and again about one of the vertices of the quadrilateral. Ask the student to describe the similarities and differences between the two new images. Model for the student a clear and complete explanation of the sequence of similarity transformations used to map one figure onto the other. Provide the student with examples of pairs of similar figures and have the student identify the sequence of similarity transformations that map one figure onto the other. Remind the student to be as clear and concise as possible in the description, identifying specifically the center and scale factor of the dilation and the distance and direction of the translation. If possible, have the student ask another student to read his or her description to see if it can be followed without further explanation. Remind the student to always include the vertices that are mapped onto corresponding vertices by the transformation in his or her explanation.
Consider implementing MFAS task To Be or Not To Be Similar (GSRT.1.2) and/or The Consequences of Similarity (GSRT.1.2). 
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 the quadrilaterals are similar and justifies this by describing a sequence of similarity transformations that carries one quadrilateral onto the other. For example, the student describes the following sequence:
 Dilate quadrilateral ABCD about the origin using a scale factor of so that (0, 1), (1, 1), (4, 1) and (5, 2).
 Translate quadrilateral six units left on the horizontal axis and two units up on the vertical axis (or according to a corresponding vector that is shown) so that (6, 1), (5, 3), (2, 3) and (1, 0).
This carries quadrilateral to quadrilateral EFGH with vertices E (6, 1), F (5, 3), G (2, 3) and H (1, 0). Therefore, quadrilateral ABCD is similar to quadrilateral EFGH.
Note: The student could also choose to dilate quadrilateral ABCD about a point other than the origin.

Questions Eliciting Thinking Could you have dilated the image about a different point? How?
What is the difference between dilating about the origin and dilating about a point not on the origin? How are the resulting figures the same? How are they different?
Is it always necessary to include a dilation when describing the sequence of similarity transformations used to verify two figures are similar? Why or why not?
Does it matter which transformation is done first? 
Instructional Implications Challenge the student with similar tasks that require more than two transformations in order to show that one figure is similar to another.
Consider implementing MFAS task To Be or Not To Be Similar (GSRT.1.2) and/or The Consequences of Similarity (GSRT.1.2). 