Download Proving a Rectangle Is a Parallelogram

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 60826
Proving a Rectangle Is a Parallelogram
Students are asked to prove that a rectangle is a parallelogram.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, proof, rectangle, parallelogram
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProvingARectangleIsAParallelogram__Worksheet.docx
MFAS_ProvingARectangleIsAParallelogram__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 Proving a Rectangle Is a Parallelogram 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.
Makes some observations about rectangles without regard to the statement to be proven.
page 1 of 4 Uses the theorem that is to be proven in its own proof.
Assumes the statement to be proven and then makes some deductions based on this statement.
Questions Eliciting Thinking
What do you know about this figure? What are you trying to prove?
What is the definition of a rectangle? What is the definition of a parallelogram?
Did you think of a plan for your proof before you started?
Instructional Implications
Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall
strategy used in the proof.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first determine 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., W Y ) and guide the student to use 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 understands that showing opposite sides parallel will lead to the conclusion that the rectangle is a parallelogram. However, the student does not adequately
show opposite sides parallel.
Questions Eliciting Thinking
What is your general strategy for this proof?
What makes a figure a parallelogram? What do you know about rectangles that might help you show this?
How can you show that opposite sides of the rectangle are parallel? What do you know about the angles in a rectangle?
Instructional Implications
Assist the student in clarifying his or her strategy. Then, review theorems that can be used to prove two lines are parallel, and guide the student to use these theorems to
complete the proof.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., W Y ) and guide the student to use notation appropriately.
page 2 of 4 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 angles in the rectangle are all right angles.
Pairs of angles are supplementary.
Both pairs of opposite sides are parallel.
Questions Eliciting Thinking
What must be true for angles to be supplementary? Did you show this in your proof?
What must be true for two lines to be parallel? Did you show this in your proof?
What must be true for this figure to be a parallelogram? Did you show this in your proof?
Instructional Implications
Provide feedback to the student concerning specific errors or omissions in his or her proof. Allow the student to revise the proof. Give the student additional opportunities
to prove statements about parallelograms and special quadrilaterals. Pair the student with another Almost There student and have the students compare proofs and
reconcile any differences.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., W Y ) 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 Parallelogram Diagonals Bisect
(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 provides a complete proof with appropriate justification such as:
Since WXYZ is a rectangle then
W,
X,
Y and
Z are right angles so that m
Z = 180° so that by definition of supplementary angles, and
W and
W =m
X=m
X are supplementary as are
Y =m
W and
Z = 90°. Then, m
W +m
X = 180° and m
W +m
Z. Since the same side interior angles are supplementary then
. So by definition WXYZ is a parallelogram.
Questions Eliciting Thinking
Is there another method you could have used to prove that a rectangle is a parallelogram?
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 proofs.
Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Angle Congruence (G-CO.3.11), Proving Parallelogram Diagonals Bisect
(G-CO.3.11) or Proving Congruent Diagonals (G-CO.3.11) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Proving a Rectangle Is a Parallelogram 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
Description
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include:
page 3 of 4 MAFS.912.G-CO.3.11:
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