Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 60825
Proving Parallelogram Diagonals Bisect
Students are asked to prove that the diagonals of a parallelogram bisect each other.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, proof, parallelogram, diagonals, bisect
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProvingParallelogramDiagonalsBisect_Worksheet.docx
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 proof on the Proving Parallelogram Diagonals Bisect worksheet.
2. The teacher asks follow-up questions, as needed.
Note: Remind the student that a theorem cannot be used in its own proof. If necessary, review the theorems that have been proved prior to the introduction of this
theorem so that the student understands which theorems can be used in this proof. Also, proofs may vary from what is shown in this rubric depending on the theorems
proved prior to task implementation.
TASK RUBRIC
Getting Started
Misconception/Error
The student’s proof shows no evidence of an overall strategy or logical flow.
Examples of Student Work at this Level
The student:
States the given information, but is unable to go any further.
Uses the theorem to be proven in its own proof
page 1 of 4 Makes some observations about the parallelogram, but is unable to go any further.
Questions Eliciting Thinking
What do you know about this figure? What are you trying to prove?
Did you think of a plan for your proof before you started?
What does bisect mean? What specifically do you need to show?
Instructional Implications
Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Ask the student why each step is necessary. Then
have the student analyze and describe the overall strategy used in the proof.
Emphasize that a theorem cannot be used as justification in its own proof. Encourage the student to first question what is available to use in a proof of a particular
statement.
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using
each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be shown to be congruent.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., XY) and guide the student to use the notation appropriately
Moving Forward
Misconception/Error
The student’s proof shows evidence of an overall strategy but fails to establish major conditions leading to the prove statement.
Examples of Student Work at this Level
The student attempts to prove
(or
) in order to show that corresponding parts of congruent triangles are congruent but is unable to
show the triangles are congruent.
Questions Eliciting Thinking
What is your general strategy for this proof?
How can you show two triangles are congruent?
What do you need to show in order to use the ASA congruence theorem? Have you done this in your proof?
Suppose you show the triangles congruent. How will you prove that the diagonals of the rectangle bisect each other?
Instructional Implications
Encourage the student to begin the proof process by developing an overall strategy. Provide another statement to be proven and have the student compare strategies
with another student and collaborate on completing the proof.
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using
each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be shown to be congruent.
page 2 of 4 If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., XY) and guide the student to use the notation appropriately.
Almost There
Misconception/Error
The student fails to establish a condition that is necessary for a later statement.
Examples of Student Work at this Level
The student fails to establish:
The congruence of one pair of angles or sides necessary to use the congruence theorem cited.
The opposite sides of the parallelogram are parallel.
Questions Eliciting Thinking
What must be true for alternate interior angles to be congruent?
I see that you stated these triangles are congruent. Can you show me all of the steps needed to use the theorem that you used? Did you use them in your proof?
Instructional Implications
Using colored pencils or highlighters, encourage the student to mark the statements that support the congruence theorem chosen. 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 ASA to prove the triangles congruent, the proof must include showing the
congruence of two pairs of corresponding angles and their included sides (and a reason or justification must be provided for each).
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., XY) and guide the student to use the notation appropriately.
Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Angle Congruence (G-CO.3.11), Proving a Rectangle Is a Parallelogram
(G-CO.3.11) or Proving Congruent Diagonals (G-CO.3.11) if not previously used.
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 presents a complete and convincing proof that the diagonals of a parallelogram bisect each other.
page 3 of 4 Questions Eliciting Thinking
Were there any statements in your proof that you did not really need? Could you have shown the diagonals bisect each other by showing that just one pair of triangles is
congruent?
Is there another method you could have used to prove that the diagonals bisect each other?
Instructional Implications
Challenge the student to prove other statements about parallelograms, squares, rectangles and rhombi. Provide the student opportunities to write proofs using a variety of
formats some of which include narrative paragraphs, flow diagrams, and two-column format.
Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Angle Congruence (G-CO.3.11), Proving a Rectangle Is a Parallelogram
(G-CO.3.11) or Proving Congruent Diagonals (G-CO.3.11) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Proving Parallelogram Diagonals Bisect 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.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.
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