Download Midpoints of Sides of a Quadrilateral

Midpoints of Sides of a Quadrilateral
Resource ID#: 59184
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are asked to prove that the quadrilateral formed by connecting the midpoints of the sides of
a given quadrilateral is a parallelogram.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, quadrilateral, coordinates, special quadrilateral, midpoints, parallelogram
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_MidpointsOfSidesOfaQuadrilateral_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 problem on the Midpoints of Sides of a
Quadrilateral worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not have an effective strategy for completing the proof.
Examples of Student Work at this Level
The student calculates (correctly or incorrectly) the coordinates of the midpoints of the sides (E, F, G and H)
Questions Eliciting Thinking
What are you asked to show in this problem?
How can you show a quadrilateral is a parallelogram?
Instructional Implications
Review the conditions that are necessary and sufficient for a quadrilateral to be a parallelogram, i.e., opposite
slope formula to show that segments are parallel, the distance formula to show that segments are congruent, a
Guide the student to develop an overall strategy for solving the problem presented in this task, i.e., (1) find th
and provide feedback.
Give the student additional opportunities to use the slope, distance, and midpoint formulas in a variety of prob




Two segments having the same slope.
Two segments having opposite reciprocal slopes.
Two segments having the same length.
The two diagonals of a quadrilateral having the same midpoint.
If needed, provide feedback on the appropriate use of notation.
Moving Forward
Misconception/Error
The student does not explicitly draw an appropriate conclusion to complete the proof.
Examples of Student Work at this Level
The student correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) and provides wo
Questions Eliciting Thinking
Can you explain how you showed the figure is a parallelogram?
What is your conclusion? Is the quadrilateral a parallelogram?
Instructional Implications
Discuss with the student how to write a clear and complete proof. Show the student a model coordinate geom
explicitly stated, and no extraneous work is left on the paper).
If needed, provide feedback on the appropriate use of notation.
Almost There
Misconception/Error
The student does not use mathematical terminology or notation correctly.
Examples of Student Work at this Level
The student correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) ,shows appropri
Uses notation incorrectly, e.g., refers to
as EH.
Uses the term congruent to describe slopes that are equal.
Questions Eliciting Thinking
What is the difference between
and EH?
What does congruent mean? Can slopes be congruent?
Instructional Implications
Provide direct feedback to the student regarding his or her use of notation. Review the use of notation with re
notation has been used incorrectly and to rewrite the statements so that they are written correctly.
Consider implementing MFAS tasks Describe the Quadrilateral (G-GPE.2.4), Diagonals of a Rectangle (G-G
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 correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) ,shows appropri
Questions Eliciting Thinking
Do you think joining the midpoints of any quadrilateral will create a figure that is a parallelogram? How coul
What if the quadrilateral is concave? Will the figure formed by connecting the midpoints of the sides be a par
Instructional Implications
Challenge the student to prove geometric theorems using coordinate geometry. Consider implementing MFAS
Consider implementing other MFAS quadrilateral tasks Diagonals of a Rectangle (G-GPE.2.4) and Describe
ACCOMMODATIONS & RECOMMENDATIONS

Special Materials Needed:
o
Midpoints of Sides of a Quadrilateral 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-GPE.2.4:
Description
Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given
points in the coordinate plane is a rectangle; prove or disprove that
the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
Remarks/Examples:
Geometry - Fluency Recommendations
Fluency with the use of coordinates to establish geometric
results, calculate length and angle, and use geometric
representations as a modeling tool are some of the most
valuable tools in mathematics and related fields.