Getting Started 
Misconception/Error The student does not understand the concept of a dilation. 
Examples of Student Work at this Level The student:
 Translates points A and B two units in some direction rather than dilates with respect to center C.
 Draws a segment containing C that is parallel to and twice as long.
 Draws a segment two units from C that is parallel to and twice as long.
 Draws a segment that is parallel to and two units longer.

Questions Eliciting Thinking What is a dilation? What is meant by the center of dilation? What is meant by the dilation factor?
How did you locate points A', B', and C'?
How is a dilation different from a translation? A reflection? 
Instructional Implications Review the definition of dilation emphasizing that a dilation maps each point, P, to a point, P', on a ray whose endpoint is the center of dilation, O, and containing P. Be sure the student understands the role of both the center and the scale factor in performing dilations (i.e., where k is the scale factor of the dilation). 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.
Repeat the task using a scale factor of 0.5 rather than two.
Provide additional experience with dilations of lines and line segments using centers that are both on and not on the lines. 
Moving Forward 
Misconception/Error The student makes errors in locating the dilated points. 
Examples of Student Work at this Level The student demonstrates an understanding of dilations, but:
 Uses the wrong point as the center of the dilation.
 Uses an incorrect dilation factor when locating one or both of the endpoints.

Questions Eliciting Thinking What is the role of the center of dilation?
What does the scale factor mean? What is its value here? How is the scale factor used in the dilation? 
Instructional Implications Remind the student that the dilated points A' and B' are measured from the center C and are a distance equal to two times their original distances from the center C. Ask the student to reconsider the locations of A' and B' and allow the student to revise his or her work. Discuss with the student what happens to following the dilation process and guide the student to compare the image of to its preimage. Be sure the student understands that is parallel to and that is twice as long as .
Provide additional experience with dilations of lines and line segments using centers that are both on and not on the lines. 
Almost There 
Misconception/Error The student is unable to provide a general description of the relationship between a line segment and its image after a dilation. 
Examples of Student Work at this Level The student correctly dilates points A and B but is unable to describe, in general, the relationship between and . For example, the student:
 States that the segment and its image are contained in parallel lines or that the image is twice the length as its preimage (but not both).
 Incorrectly describes the relationship between a length (CA) and the location of a point (A’).
 Refers to a segment as a line.

Questions Eliciting Thinking What happened to after you dilated it? Where was it located in relationship to its original location?
How does the length of compare to the length of ?
What did you mean by A’= 2(CA)? Can a point be equal to a number? How could you correct this statement?
What is the difference between a line and a line segment? Is a line or a line segment? 
Instructional Implications Provide feedback to the student concerning any misuse of terminology, notation, or omissions in his or her description. Ask the student to address both the orientation of the segment and its image (i.e., the parallel relationship) and a comparison of their lengths.
Provide additional experience with dilations of lines and line segments using centers that are both on and not on the lines. 
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 graphs the images of points A and B after a dilation with center at point C and scale factor of two and draws . The student explains that and are contained in parallel lines and that is twice the length of . 
Questions Eliciting Thinking If the scale factor is r, how will the lengths of the segment and its image compare in terms of r?
In general, what is the relationship between a segment and it image after a dilation with center at point C and using a scale factor of r? 
Instructional Implications Ask the student to describe the image of an equilateral triangle whose sides are dilated by a scale factor of r using one of the vertices of the triangle as the center of dilation. 