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The Consequences of Similarity
Resource ID#: 64452
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are given the definition of similarity in terms of similarity transformations and are
asked to explain how this definition ensures the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, similar, transformations, similarity transformations, triangles
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TheConsequencesOfSimilarity_Worksheet.docx
MFAS_TheConsequencesOfSimilarity_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with a small group, or with the whole class.
1. The teacher asks the student to complete the problems on the Consequences of Similarity
worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand similarity in terms of transformations.
Examples of Student Work at this Level
The student:

Writes the corresponding angles are congruent and the corresponding sides are
proportional because the triangles are similar.

Explains in terms of the converse of the statement (i.e., if corresponding angles are
congruent, then the triangles are similar).



Writes that without a diagram, he or she cannot determine whether or not the
corresponding angles are congruent and/or the corresponding sides are proportional.
Writes that both the corresponding angles and the corresponding sides would be
proportional.
Indicates that he or she does not understand similarity in terms of transformations.
Questions Eliciting Thinking
What does it mean for two triangles to be similar?
How is similarity determined or verified?
How can transformations be used to define and justify similarity?
How can you determine, based on a sequence of similarity transformations, that
?
Do similarity transformations preserve angle measure?
In a dilation, are the corresponding sides proportional?
What is scale factor, and how does it relate to similarity?
Instructional Implications
Review the definition of similarity in terms of similarity transformations. Explain that two
triangles are similar if there is a dilation or a dilation and a congruence (i.e., a sequence of
rigid motions) which carries one triangle onto the other. Continue to explain that a dilation
changes the lengths of the sides by a given scale factor, r, such that
and
therefore, ensures proportionality of corresponding sides. A dilation also preserves angle
measure and therefore, ensures corresponding angles are congruent. Remind the student that
rigid motions preserve lines, rays and segments and are both angle- and distance-preserving. If
the student is still struggling with his or her understanding of similarity in terms of similarity
transformations, provide the student with several examples of similar triangles. Have the
student identify the composition of transformations that maps one triangle onto the other. Lead
the student to understand that because this composition of similarity transformations exists, the
triangles are similar. Furthermore, because the triangles are similar, we know that there must
be a composition of transformations that maps one triangle onto the other. Because similarity
transformations preserve angle measures, we know that corresponding angles are congruent,
and because dilations change side length by a given scale factor, we know that corresponding
sides are proportional.
Consider implementing one of the following MFAS tasks Showing Similarity (G-SRT.1.2), To
Be or Not To Be Similar (G-SRT.1.2).
If the student does not have a clear understanding of dilations, 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. Review with the student the Fundamental Theorem of Similarity, which ensures
that the sides of the triangle not lying on the center of dilation are parallel to the corresponding
sides. The corresponding angles are therefore congruent because corresponding angles of
parallel lines are congruent.
Making Progress
Misconception/Error
The student understands the definition of similarity in terms of similarity transformations but
omits important details or does not completely describe how this definition ensures the
equality of all corresponding pairs of angles and the proportionality of all corresponding pairs
of sides.
Examples of Student Work at this Level
The student attempts to reason from similarity transformations to the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
However, the student does not directly or explicitly refer to similarity transformations (i.e.,
rigid motion and dilations) and their properties to explain why angle measure is preserved and
lengths of sides are proportional.
Questions Eliciting Thinking
What is the definition of similarity in terms of similarity transformations?
What properties are preserved in translations, rotations, and reflections?
What properties are preserved in dilations?
Can you describe in more detail how you know the corresponding angle measures are equal?
Can you describe in more detail how you know the corresponding side lengths are
proportional?
Instructional Implications
Model for the student a clear and complete explanation of why the definition of similarity in
terms of similarity transformations ensures the equality of all corresponding pairs of angles
and the proportionality of all corresponding pairs of sides. Explain that if
, there
is a composition of rigid motion and dilation that maps
to
. Since both rigid
motion and dilation preserve angle measure, the angles of
must be congruent to the
corresponding angles of
. Also, a dilation changes the lengths of the sides by a given
scale factor, r, such that
, and therefore,
which ensures proportionality of corresponding sides.
Consider implementing MFAS task Congruence Implies Congruent Corresponding Parts (GCO.2.7) to reinforce the student’s understanding of rigid motions ensuring congruent
corresponding parts.
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 writes:
1. Since
, there is a composition of rigid motion and dilation that maps
to
. Since both rigid motion and dilation preserve angle measure, the
angles of
must be congruent to the corresponding angles of
.
2. Since dilations change lengths by a scale factor given by the dilation, there is some
scale factor, r, such that AB = r(DE), BC = r(EF), and AC = r(DF). Consequently,
AB:DE as BC:EF as AC:DF so that corresponding sides are proportional.
Questions Eliciting Thinking
How are the definitions of congruence and similarity (in terms of transformations) alike, and
how are they different?
Is there a sequence of dilations, or dilations and rigid motions that would produce an image
that is not similar to the preimage? Why or why not?
Instructional Implications
Review/explain the Triangle Midsegment Theorem to the student. Ask the student to verify,
using his or her understanding of a dilation, why the midsegment is parallel to and half the
length of the third side of the triangle.
Consider implementing MFAS task Dilation of a Line Segment (G-SRT.1.1).
ACCOMMODATIONS & RECOMMENDATIONS

Special Materials Needed:
o
The Consequences of Similarity 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
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.1.2:
Description
Given two figures, use the definition of similarity in terms of
similarity transformations to decide if they are similar; explain
using similarity transformations the meaning of similarity for
triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.