Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70737
Explaining a Proof of the Pythagorean Theorem
Students are asked to explain the steps of a proof of the Pythagorean Theorem that uses similar triangles.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, Pythagorean Theorem, proof, similar triangles, properties
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ExplainingAProofOfThePythagoreanTheorem_Worksheet.docx
MFAS_ExplainingAProofOfThePythagoreanTheorem_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 Explaining a Proof 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 of the Pythagorean Theorem. The student:
Is unable to recognize or justify the similar triangles.
page 1 of 4 Does not relate the proof to the Pythagorean Theorem.
Makes errors in justifying the steps of the proof with appropriate properties or theorems.
Provides answers too brief to demonstrate understanding of the proof.
Questions Eliciting Thinking
What do the triangles have in common that can be used to justify their similarity?
Why is it significant that
is perpendicular to
?
What is the given information? What assumptions can you make knowing that
is a right triangle?
What are you trying to prove? What does the Pythagorean Theorem state?
What does the Substitution Property (or other property) state? How can you justify the substitution (or other action)?
Instructional Implications
Provide the student with basic instruction on the Pythagorean Theorem. 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 Pythagorean Theorem, explicitly state the
assumptions (a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c) and the conclusion
Describe the Pythagorean Theorem
both verbally and in symbols and show applications of its use.
As needed, review the properties and theorems used in the proof: Angle-Angle Similarity Theorem, Distributive Property, Addition Property of Equality, Substitution Property
and Cross Products Property of Proportions. Make clear what each property states and provide examples to demonstrate the application of each property.
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 Pythagorean Theorem modeling the use of definitions, properties, or theorems to justify each step of the proof.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., AD) 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 is able to reference the properties applied in the proof, but does not fully justify or explain one or more steps of the proof. The student:
Does not reference the relationship among
,
,and
to justify the substitution in step 6.
page 2 of 4 Explains step 7 only as proving the Pythagorean Theorem.
Makes a minor mistake in an explanation.
Questions Eliciting Thinking
How do you know c = d + e?
Can you explain what you mean by “it proves the Pythagorean theorem”? What specifically was proven?
What do you mean by the sides are “similar”? Can lengths of sides be similar?
Instructional Implications
Provide the student with feedback on his or her answers and prompt the student to supply justifications that are missing. If necessary, review notation for naming sides,
lengths of sides, angles, and angle measures. Also, review that when naming similar triangles, vertices are named in corresponding order.
Consider implementing MFAS task Converse of the Pythagorean Theorem (8.G.2.6) or MFAS tasks from standard 8.G.2.5 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. The Angle-Angle Similarity Theorem proves
because
and
. Also,
because
and
.
2. The corresponding sides of similar triangles are proportional by the definition of similar triangles.
3. The Cross Products Property of Proportions was applied to each proportion.
4. The Addition Property of Equality was used. Since
and
,
can be added to
and cd can be added to ce.
5. The Distributive Property was used to rewrite the expression cd + ce as c(d + e).
6. Because c = d + e (i.e., AD + DB = AB) by the Segment Addition Postulate, c can be substituted for d + e so that
.
7. You have proven that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse which proves the
Pythagorean Theorem.
Questions Eliciting Thinking
Can you explain the proof in your own words?
Why is it important to justify each step of a proof?
Instructional Implications
Provide opportunities to use the Pythagorean Theorem to find missing lengths in right triangles.
Consider implementing MFAS task Converse of the Pythagorean Theorem (8.G.2.6) or MFAS tasks from standard 8.G.2.5 if not used previously.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
page 3 of 4 Explaining a Proof 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
Name
MAFS.8.G.2.6:
Description
Explain a proof of the Pythagorean Theorem and its converse.
page 4 of 4