Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 123108
An Isosceles Trapezoid Problem
Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Keywords: MFAS, trapezoid, isoceles trapezoid, diagonals, perimeter
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_AnIsoscelesTrapezoidProblem_Worksheet.docx
MFAS_AnIsoscelesTrapezoidProblem_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 An Isosceles Trapezoid Problem worksheet.
2. The teacher asks follow-up questions, as needed.
Note: In the rubric, the following properties and theorems are referenced by name:
Addition Property of Inequality – For all real numbers a, b, and c, if a > b, then a + c > b + c.
Diagonals of an Isosceles Trapezoid Theorem – The diagonals of an isosceles trapezoid are congruent.
Triangle Inequality Theorem – The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand the statement that needs to be explained.
Examples of Student Work at this Level
The student does not make explicit that the objective is explain why RP + T A < RT + T P + PA + AR. The student may write some true and relevant statements on his or
her paper but never makes clear that RP + TA must be shown to be less than RT + T P + PA + AR.
page 1 of 4 Questions Eliciting Thinking
Can you explain in words what you are trying to show?
Can you write an expression that represents the perimeter of this trapezoid? Can you write an expression that represents the sum of the lengths of the diagonals of the
trapezoid? Specifically what are you trying to show in this problem?
How will the statements that you wrote help you to explain why RP + T A < RT + T P + PA + AR?
Instructional Implications
Guide the student to understand that in order to explain why the sum of the lengths of the diagonals of TRAP is less than its perimeter, it is necessary to show that RP +
T A < RT + T P + PA + AR. Be sure the student understands how to represent both the perimeter of the trapezoid and the sum of the lengths of its diagonals. Review the
Triangle Inequality Theorem, the Diagonals of an Isosceles Trapezoid Theorem, and the Addition Property of Inequality. Then guide the student through the statements of
an explanation or proof and prompt the student to provide justifications.
Address any misuse of notation and allow the student to make corrections on his or her paper.
Provide additional opportunities to explain or prove statements about triangles. Consider implementing other MFAS tasks aligned to standard G-CO.3.10.
Moving Forward
Misconception/Error
The student understands the statement to be explained but is unable to write a complete and coherent explanation.
Examples of Student Work at this Level
The student understands the need to show that RP + T A < RT + T P + PA + AR. However, the student is unable to write a completely correct explanation or proof. For
example, the student:
Writes a statement incorrectly which impacts the remainder of the explanation or proof.
Omits important statements or justifications.
Is unable to complete the explanation or proof.
Questions Eliciting Thinking
What are you trying to show in this problem?
How will the statements you wrote help you explain why RP + T A < RT + T P + PA + AR?
Is your explanation complete? What part were you unable to complete?
Instructional Implications
Provide feedback to the student concerning any incorrectly written statements and ask the student to revise his or her explanation. Prompt the student to provide any
missing justifications. If needed, assist the student in completing his or her explanation.
Address any misuses of notation and allow the student to make corrections on his or her paper.
page 2 of 4 Provide additional opportunities to explain or prove statements about triangles. Consider implementing other MFAS tasks aligned to standard G-CO.3.10.
Almost There
Misconception/Error
The student’s explanation includes a minor error.
Examples of Student Work at this Level
The student writes an essentially correct explanation. However, the student:
Uses notation incorrectly.
Does not make clear how the inequality RP + T A < RT + T P + PA + AR relates to the statement to be explained.
Questions Eliciting Thinking
How should a length be represented? How is this different from naming a segment?
Why does showing that RP + T A < RT + T P + PA + AR explain why the sum of the lengths of the diagonals of TRAP is less than its perimeter?
Instructional Implications
Review conventions in naming segments and their lengths and ask the student to make any needed corrections to notation. Guide the student to write statements that
make explicit RP + T A represents the sum of the lengths of the diagonals and RT + T P + PA + AR represents the perimeter of the trapezoid so showing that RP + T A < RT
+ T P + PA + AR proves the sum of the lengths of the diagonals is less than the perimeter of the trapezoid.
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 and correct explanation of the statement that the sum of the lengths of the diagonals of TRAP is less than its perimeter. For example, the
student writes:
In triangle TRP, RP < RT + T P and in triangle RAP, RP < PA + AR (by the Triangle Inequality Theorem). Therefore, RP + RP < RT + T P + PA + AR (by the Addition
Property of Inequality). Since trapezoid TRAP is isosceles, RP = T A (by the Diagonals of an Isosceles Trapezoid Theorem) so that RP + T A < RT + T P + PA + AR. (by
substitution). Since the sum of the lengths of the diagonals is given by T A + RP and the perimeter of TRAP is given by RT + T P + PA + AR, the sum of the lengths of the
diagonals of TRAP is less than the perimeter of TRAP.
Questions Eliciting Thinking
Where in your explanation did you rely on the fact that the trapezoid is isosceles?
Do you think the trapezoid has to be isosceles in order for its perimeter to be greater than the sum of the lengths of its diagonals?
Instructional Implications
Challenge the student to determine if the sum of the lengths of the diagonals of any trapezoid is less than its perimeter. If so, ask the student to write a proof; if not, ask
the student to explain why not.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
An Isosceles Trapezoid Problem worksheet
SOURCE AND ACCESS INFORMATION
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
page 3 of 4 Name
MAFS.912.G-CO.3.10:
Description
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of
interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
page 4 of 4