Getting Started 
Misconception/Error The student does not understand the concept of a dilation. 
Examples of Student Work at this Level The student:
 Reflects point A and B across point C on .
 Translates points A and B 1.5 units in some direction rather than dilates with respect to center C.

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' and B'?
How is a dilation different from a translation (or 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. Discuss 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 2.5 rather than 1.5.
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:
 Locates the dilated points, A' and B', by measuring from A and B, rather than from the center C.
 Uses an incorrect dilation factor.

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 1.5 times their original distances from the center C. Ask the student to reconsider the locations of A' and B' and then explain where to locate C'. If necessary, prompt the student to indicate how far C' should be from C. 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 the line containing points A' and B' is the same as the line containing points A and B. Repeat the task using a scale factor of 2.5 rather than 1.5.
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 and its image after dilating about a center on the line. 
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 a line and its image after dilating about a center on the line. For example, the student says:
 The line is still a line.
 The line gets longer.

Questions Eliciting Thinking What happened to after you dilated it? Where was it located in relationship to its original location?
What is the difference between a line and a line segment?
Does a line have length? How long is a line? What is the length of ? 
Instructional Implications Review the distinction between a line and a line segment. Acknowledge that the locations of points A and B changed as a result of the dilation but so would the locations of all points on the line (other than the center of dilation, C). Explain that the line still contains infinitely many points. Guide the student to observe that the line containing points A' and B' is the same as the line containing points A and B.
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 correctly graphs the images of points A and B after a dilation with center at point C and scale factor of 1.5. The student says the image of is the same as . The student explains that, in general, the image of a line after a dilation with a center on the line will always be the same line.

Questions Eliciting Thinking You said the image of a line after a dilation with a center on the line will always be the same line. Is this true regardless of the scale factor?
How would you describe the result of a dilation of ? 
Instructional Implications Ask the student to explore and describe:
 The dilation of using a center that is not on the line.
 The dilation of using centers both on and not on .
