Download Congruence Implies Congruent Corresponding Parts

Primary Type: Formative Assessment
Status: Published
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Resource ID#: 59703
Congruence Implies Congruent Corresponding Parts
Students are given two congruent triangles and asked to determine the corresponding side lengths and angle measures and to use the definition of
congruence in terms of rigid motion to justify their reasoning.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, congruence, congruent pairs, triangle, transformation, rigid motion, translation, rotation,
reflection
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_CongruenceImpliesCongruentCorrespondingParts_Worksheet.pdf
MFAS_CongruenceImpliesCongruentCorrespondingParts_Worksheet.docx
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 Congruence Implies Congruent Corresponding Parts worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
Given a congruence statement, the student is unable to identify corresponding sides and/or angles.
Examples of Student Work at this Level
The student writes something like:
.
DF = 4.
Questions Eliciting Thinking
page 1 of 3 Can you explain how you determined your answers?
If you were not shown the triangles, how could you determine which vertices correspond from the congruence statement?
What are corresponding sides and angles? Can you identify the corresponding sides and angles of the two congruent triangles?
Instructional Implications
Explain to the student that the congruence statement reflects the correspondence between the vertices of the two triangles, so pairs of corresponding angles and
corresponding sides can be identified. Show the student how to use the congruence statement to determine the six pairs of corresponding parts and how to write the six
congruence statements that result. Be sure the student uses appropriate notation. Provide the student with examples of pairs of congruent triangles in which
corresponding parts are marked congruent. Ask the student to write a congruence statement for each.
Address issues with the use of the definition of congruence in terms of rigid motion to justify the congruence of corresponding parts (see Moving Forward Instructional
Implications).
Moving Forward
Misconception/Error
The student does not understand that the congruence ensures there is a sequence of rigid motions that will map one triangle onto the other.
Examples of Student Work at this Level
After identifying the lengths of the sides and measures of the angles of
, the student attempts to describe a sequence of rigid motions that maps one triangle onto
the other (either correctly or incorrectly).
Questions Eliciting Thinking
What is the definition of congruence in terms of rigid motion?
If two triangles are congruent, what does the definition tell you must be true?
Instructional Implications
Explain to the student that this task is a part of the proof of the statement, “Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs
of angles are congruent.” Explain that when proving a statement of the form “p if and only if q,” there are actually two statements that must be established: “If p then q,”
and “If q then p.” Consequently, to prove the statement, “Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are
congruent,” one must prove both of the following: “If two triangles are congruent, then corresponding pairs of sides and corresponding pairs of angles are congruent,” and
“If, given two triangles, corresponding pairs of sides and corresponding pairs of angles are congruent, then the two triangles are congruent.” The problem in this task is
addressing the former statement.
Review the definition of congruence in terms of rigid motion and explain to the student that the congruence of the two triangles ensures there is a sequence of rigid
motions that will map one triangle onto the other, so it is not necessary to identify the sequence. Review the basic properties of rigid motion, [i.e., (1) rigid motions map
lines to lines, rays to rays, and segments to segments, (2) rigid motions are distance preserving, and (3) rigid motions are degree preserving]. Assist the student in
understanding that since ABC
DEF, there is a sequence of rigid motions that maps ABC onto DEF. Since rigid motions preserve both distance and angle
measure, then
,
,
,
,
, and
, so the corresponding parts of the congruent triangles are congruent.
Allow the student ample time to explore how sequences of rigid motions ensure congruence by using manipulatives or an interactive website (e.g.,
http://nlvm.usu.edu/en/nav/frames_asid_294_g_4_t_3.html?open=activities&from=category_g_4_t_3.html). Have the student use the website (or manipulatives and graph
paper) to model a sequence of rigid motions that carries one figure onto another. Ask the student to describe how the basic properties of rigid motion are demonstrated in
this activity. Make sure the student understands that because rigid motion preserves both distance and angle measure, the corresponding parts of the preimage and image
are congruent.
Be sure to address any issues with the misuse of notation (e.g., writing
= 6 rather than DE = 6 or
rather than
).
Consider implementing MFAS task Showing Congruence Using Corresponding Parts - 1 (G­CO.2.7) which addresses the proof of the statement, “If, given two triangles,
corresponding pairs of sides and corresponding pairs of angles are congruent, then the two triangles are congruent."
Almost There
Misconception/Error
The student is unable to generalize from the example to all triangles.
Examples of Student Work at this Level
page 2 of 3 The student is unable to clearly explain how the reasoning used in question 2 can be applied to any pair of triangles.
Questions Eliciting Thinking
If I changed the way these two triangles look or are oriented on the paper, how could you determine the corresponding pairs of parts that are congruent?
How could you apply your reasoning in this example to any pair of triangles?
Instructional Implications
Explain to the student that there is nothing about the reasoning used to conclude
and
,
,
,
,
, and
that is specific to
. This reasoning can be applied to any pair of triangles. Alter the triangles given in the diagram and ask the student to complete questions 1 and 2 again.
Then ask the student to compare responses to question 2 for the two versions of the exercise.
Be sure to address any issues with the misuse of notation (e.g., writing
= 6 rather than DE = 6 or
rather than
).
Consider implementing MFAS task Showing Congruence Using Corresponding Parts - 1 (G­CO.2.7) which addresses the proof of the statement, “If, given two triangles,
corresponding pairs of sides and corresponding pairs of angles are congruent, then the two triangles are congruent.”
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 determines the lengths of the sides of
F=
. The student reasons that if
angle measure, then
to
ABC and
ABC
,
DEF are DE = 6, DF = 5, and EF = 4, and the measures of the angles of
DEF, then there is a sequence of rigid motions that maps
,
,
,
,
ABC onto
DEF are m
D=
,m
E=
, and m
DEF. Since rigid motions preserve both distance and
. The student further explains there is nothing about this reasoning that is specific
DEF; this reasoning can be applied to any pair of triangles. So, if two triangles are congruent, then each pair of corresponding sides and each pair of
corresponding angles are congruent.
Questions Eliciting Thinking
You showed that if two triangles are congruent, then corresponding pairs of sides and corresponding pairs of angles are congruent. What is the converse of this statement?
What must be done to prove the converse?
Instructional Implications
Consider implementing MFAS task Showing Congruence Using Corresponding Parts - 1 (G­CO.2.7) which addresses the proof of the statement, “If, given two triangles,
corresponding pairs of sides and corresponding pairs of angles are congruent, then the two triangles are congruent.”
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Congruence Implies Congruent Corresponding Parts 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-CO.2.7:
Description
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
page 3 of 3