Download Converse of the Pythagorean Theorem

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70752
Converse of the Pythagorean Theorem
Students are asked to explain the steps of a proof of the converse of the Pythagorean Theorem.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, Pythagorean Theorem, proof, converse, properties
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ConverseOfThePythagoreanTheorem_Worksheet.docx
MFAS_ConverseOfThePythagoreanTheorem_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 problems on the Converse of the Pythagorean Theorem worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to explain and justify the steps of the proof.
Examples of Student Work at this Level
The student may be able to explain some of the steps or name some of the properties used as justifications, but the student does not demonstrate an overall
understanding of the proof. The student:
Makes assumptions that have no basis (e.g., immediately states
ABC is a right triangle).
page 1 of 4 Says substitutions can be made because the variables name the same parts of the respective triangles (e.g., both are hypotenuses).
Provides answer(s) that do not address the question asked.
Is unable to complete the proof.
Questions Eliciting Thinking
What information is given? What is being proven?
What does the converse of the Pythagorean Theorem state?
Can the substitution in the second question be justified by saying all the variables represent legs of a triangle? Why can a and b substitute for r and s? Look closely at the
given information.
Can you say
just because c and t both name a hypotenuse?
When is it appropriate to apply the Substitution Property?
Instructional Implications
Provide the student with basic instruction on the Pythagorean Theorem and its converse. Review the parts of a right triangle (e.g., vertices, right angle, acute angles,
hypotenuse, and legs) and be sure the student understands the distinction between the legs and the hypotenuse. When initially introducing the two theorems, explicitly
state the assumptions and conclusion:
Pythagorean Theorem: Assume a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c. Then prove
Converse: Assume a triangle has sides whose lengths are related by the equation
.
. Then prove the triangle is a right triangle.
Be sure the student understands the distinction between the Pythagorean Theorem and its converse.
Provide instruction on writing mathematical explanations, justifications, and proofs. Encourage the student to first consider the statement to be proven. Next, ask the
student to examine the assumptions and then formulate an overall strategy. Make clear that every step must be justified with mathematical properties or theorems. Guide
the student through the proof of the converse of the Pythagorean Theorem modeling the use of definitions, properties, or theorems to justify each step of the proof.
If necessary, review notation for naming angles (e.g.,
) and describing measures of angles (e.g.,
) and guide the student to use the notation appropriately.
Provide additional opportunities for the student to write informal proofs. Consider implementing MFAS tasks for standard 8.G.1.5 which ask the student to justify various
angle relationships.
Making Progress
Misconception/Error
The student does not clearly justify each step of the proof.
Examples of Student Work at this Level
The student demonstrates overall understanding of the proof but does not fully justify or explain one or more steps of the proof. The student:
Makes a minor mistake in an explanation.
Provides an incomplete explanation.
page 2 of 4 Questions Eliciting Thinking
What is the Substitution Property of Equality? How does the Substitution Property justify that
?
Can you explain your answer (Pythagorean Theorem) to the first question? How can you justify the use of the Pythagorean Theorem?
You said
is congruent to
. Why is that significant? What does it mean about triangle ABC?
Instructional Implications
Provide the student with feedback on the answers and prompt the student to supply justifications that are missing. Emphasize that every step must be justified using the
given information and mathematical properties or theorems. If necessary, review notation for naming sides, lengths of sides, angles, and angle measures.
Consider implementing MFAS task Explaining a Proof of the Pythagorean Theorem (8.G.2.6) or MFAS tasks from standard 8.G.1.5 if not used previously.
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 justifies each step of the proof with a complete explanation and specific evidence. For example, the student writes:
1. Mr. Lopez can use the Pythagorean Theorem because it is given that
is a right triangle.
2. It is given that a = r and b = s, so the Substitution Property can be applied.
3. Because
4. Because
Since
and
, the Substitution Property can be used to justify that
is congruent to
is equal to
.
, the sides and angles of the triangles are congruent. Therefore, you can conclude
is congruent to
which measures 90°.
measures 90°, triangle ABC is a right triangle.
Questions Eliciting Thinking
Can you explain the proof in your own words?
Why is it important to justify each step of a proof?
How can you use the converse of the Pythagorean Theorem?
Instructional Implications
Provide opportunities to use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
Consider implementing MFAS task Explaining a Proof of the Pythagorean Theorem (8.G.2.6) or MFAS tasks from standard 8.G.1.5 if not used previously.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Converse of the Pythagorean Theorem 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
page 3 of 4 Name
MAFS.8.G.2.6:
Description
Explain a proof of the Pythagorean Theorem and its converse.
page 4 of 4