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 + TA < RT + TP + 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 + TP + PA + AR.
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| 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 + TA < RT + TP + 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 + TA < RT + TP + 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 + TA < RT + TP + 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.
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| Questions Eliciting Thinking What are you trying to show in this problem?
How will the statements you wrote help you explain why RP + TA < RT + TP + 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.
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 + TA < RT + TP + PA + AR relates to the statement to be explained.
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| Questions Eliciting Thinking How should a length be represented? How is this different from naming a segment?
Why does showing that RP + TA < RT + TP + 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 + TA represents the sum of the lengths of the diagonals and RT + TP + PA + AR represents the perimeter of the trapezoid so showing that RP + TA < RT + TP + 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 + TP and in triangle RAP, RP < PA + AR (by the Triangle Inequality Theorem). Therefore, RP + RP < RT + TP + PA + AR (by the Addition Property of Inequality). Since trapezoid TRAP is isosceles, RP = TA (by the Diagonals of an Isosceles Trapezoid Theorem) so that RP + TA < RT + TP + PA + AR. (by substitution). Since the sum of the lengths of the diagonals is given by TA + RP and the perimeter of TRAP is given by RT + TP + 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. |
