Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 72961
Dilation of a Line: Factor of One Half
Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of
lines when the line does not contain the center.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, dilation, line
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_DilationOfaLIneFactorOfOneHalf_Worksheet.docx
MFAS_DilationOfaLIneFactorOfOneHalf_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Dilation of a Line: Factor of One Half worksheet.
2. The teacher asks follow-up questions, as needed.
Note: The following Theorem is referenced by name in the rubric:
The Fundamental Theorem of Similarity – Let D be a dilation with center O and scale factor r > 0. Suppose point O does not lie on
dilation D. Then
is parallel to
. Let
be the image of
under
and P’Q’ = r(PQ). For more information and a proof, see Wu (2013) which can be accessed at https://math.berkeley.edu/~wu/CCSS-
Geometry_1.pdf.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand the concept of a dilation.
Examples of Student Work at this Level
page 1 of 4 The student:
Reflects point A and B across point C on
.
Translates the points 0.5 units in some direction.
Draws a figure unrelated to the dilation.
Locates the points incorrectly with respect to the center of the dilation.
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 1.5 rather than 0.5.
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
page 2 of 4 The student demonstrates an understanding of dilations, but:
Uses the wrong center for some or all of the points.
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', B', and C' are measured from the center D and are a distance equal to 0.5 times their original distances from point D. 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
student to compare the image of
following the dilation process and guide the
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 1.5 rather than 0.5.
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 not 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:
Does not indicate that the image of
is parallel to
.
Says the line gets shorter or longer.
Confuses lines and line segments.
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
of
is parallel to
. Ask the student to describe the location
and to relate the location to the scale factor.
page 3 of 4 Provide additional experience with dilations of lines and line segments using centers that are both on and not on the lines (or lines containing the line segments).
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 0.5. Since the scale factor is 0.5,
from point D as
is from D. This places the image between D and
The student says, in general, the image of
is a line parallel to
is half as far
(since the scale factor is less than one).
at a distance from the center of dilation determined by the scale factor.
Questions Eliciting Thinking
What can you infer regarding a dilation with a scale factor less than one? How does it compare to a dilation with a scale factor greater than one?
Instructional Implications
Ask the student to repeat the problem using a scale factor of
and without the grid background.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Dilation of a Line: Factor of One Half worksheet.
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.1.1:
Description
Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing
through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
page 4 of 4