Getting Started |
| Misconception/Error The student does not understand the concept of a dilation. |
| Examples of Student Work at this Level The student:
- Connects the given points to form a triangle; then reflects and translates the triangle.
- Translates the points two units in some direction.
- Locates the points incorrectly with respect to the center of the dilation.
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| 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 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 three 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 center for some or all of the points.
- Using an incorrect dilation factor.
- Imposes a coordinate grid but with the origin at point C (instead of D).
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| 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?
If you are working in the coordinate plane, where is a convenient location for the center of the dilation? |
| Instructional Implications Remind the student that the dilated points A', B', and C' are measured from the center D and are a distance twice their original distances from point D. If the student placed the points in the coordinate plane, explain that the center of the dilation must be located at the origin in order to find the images of points by multiplying their coordinates by the scale factor, two. Ask the student to reconsider the locations of A', B', and C' and 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 the line containing points A' and B' is parallel to the line containing points A and B. Ask the student to describe the location of  and to relate the location to the scale factor. Repeat the task using a scale factor of three rather than two.
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, B, and C but is unable to describe, in general, the relationship between a line and its image after dilating about a center not on the line. For example, the student:
- Says the line gets shorter or longer.
- Confuses lines and line segments.
- Does not indicate that the image of
 is parallel to .
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| 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. Assist the student in observing that the image of is parallel to . Ask the student to describe the location of and to relate the location to the scale factor.
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, B, and C after a dilation with center at point D and scale factor of two. Since the scale factor is two, is twice as far from point D as is from D.
The student says, in general, the image of is a line parallel to at a distance from the center of dilation determined by the scale factor. |
| Questions Eliciting Thinking Why is midway between D and the image of ?
Under what conditions will lie between point D and ? |
| Instructional Implications Ask the student to generalize about the relative locations of the center of dilation, the original line, and the image of the line depending on the size of the scale factor (i.e., less than one or greater than one). |
