Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 121287
Two Congruent Triangles
Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Keywords: MFAS, prove, parallelogram, triangles, congruent
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TwoCongruentTriangles_Worksheet_.docx
MFAS_TwoCongruentTriangles_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 Two Congruent Triangles worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student’s explanation shows no evidence of the use of a strategy for showing triangles are congruent.
Examples of Student Work at this Level
There is no evidence in the student’s explanation of the use of a strategy for showing triangles are congruent (e.g., Side­Angle­Side Congruence Theorem).
page 1 of 4 The student attempts to explain using a nonexistent strategy for showing triangles are congruent (e.g., the AAA or SSA Congruence Theorems).
Questions Eliciting Thinking
Did you think through a rationale for your explanation before you started? Did you consider what you know about parallelograms that might help you explain why these
triangles are congruent?
What are some ways that you can show two triangles are congruent? What are the triangle congruence theorems?
Instructional Implications
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, and HL) and what must be established in order to conclude two triangles are congruent when
using each method. Remind the student that each letter of the theorem name represents a pair of parts that must be shown to be congruent (e.g., if using SSS to prove
the triangles congruent, the explanation must address showing three pairs of corresponding sides are congruent). Emphasize that a convincing explanation also includes
justifications of congruence statements.
Review properties of the parallelogram that might be used to solve problems about parallelograms (e.g., opposite angles of a parallelogram are congruent or opposite sides of
a parallelogram are congruent). Guide the student to apply what is known about parallel lines intersected by a transversal to a parallelogram and its diagonals.
Moving Forward
Misconception/Error
The student’s explanation shows evidence of the use of a strategy for showing triangles congruent, but the student fails to address necessary conditions leading to this
conclusion.
Examples of Student Work at this Level
The student attempts to explain that the two triangles are congruent using a triangle congruence theorem but fails to convincingly justify the pattern of corresponding
congruent parts required by the theorem.
Questions Eliciting Thinking
What do you need to show in order to use the SAS (or other relevant) Congruence Theorem?
How do you know these sides (or angles) are congruent?
Instructional Implications
Review the triangle congruence theorems and provide more opportunities and experiences with proving triangles congruent.
Review properties of the parallelogram that might be used to solve problems about parallelograms (e.g., opposite angles of a parallelogram are congruent or opposite sides of
a parallelogram are congruent). Guide the student to apply what is known about parallel lines intersected by a transversal to a parallelogram and its diagonals.
Almost There
Misconception/Error
Some aspect of the student’s explanation is incorrect or incomplete.
Examples of Student Work at this Level
The student provides an essentially correct explanation. However, the student:
Incorrectly justifies a congruence statement.
page 2 of 4 Includes unnecessary statements.
Marks congruent parts in the diagram and provides justifications but does not write a complete explanation.
Questions Eliciting Thinking
What tells you that the diagonals of a parallelogram bisect each other? Is it a definition or a theorem?
What tells you that vertical angles are congruent? Is it a definition or a theorem?
You included several unnecessary statements given that you used the SAS (or other relevant) Congruence Theorem. Can you identify the statements that are not needed
in your explanation?
Can you be more explicit in stating what pairs of parts are congruent in the diagram?
Instructional Implications
Provide feedback to the student concerning any errors, omissions, or unnecessary statements. Allow the student to revise his or her explanation. Assist the student in
understanding what a complete and correct explanation should include.
Provide opportunities to prove theorems about parallelograms that require showing 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 provides a complete explanation with justification such as:
Since the diagonals of a parallelogram bisect each other,
and
. Since vertical angles are congruent,
. So,
by the Side-Angle-
Side (SAS) Congruence Theorem.
Questions Eliciting Thinking
How do you know the diagonals of a parallelogram bisect each other?
Can you think of a second way to show the triangles are congruent?
Instructional Implications
Provide opportunities to prove theorems about parallelograms that require showing triangles are congruent.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Two Congruent Triangles worksheet
SOURCE AND ACCESS INFORMATION
page 3 of 4 Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.11:
Description
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include:
opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
page 4 of 4