Polynomials

Lesson 12

Objective

Use polynomial identities to determine Pythagorean triples.

Common Core Standards

Core Standards

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  • A.APR.C.4 — Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)2 = (x² — y²)² + (2xy)² can be used to generate Pythagorean triples.

Foundational Standards

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  • 8.G.B.7

Criteria for Success

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  1. Describe the general rule for identifying Pythagorean triples, and show examples that it works.
  2. Describe that by repeatedly identifying the same numerical relationship, you can develop an identity for a known pattern. 
  3. Deconstruct a set of Pythagorean triples into the polynomial identity.
  4. Generate Pythagorean triples using the identity.

Tips for Teachers

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Ensure that students memorize the following sets of Pythagorean triples: 3-4-5, 5-12-13, and 7-24-25. 

Anchor Problems

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Problem 1

Prove that if $${ x>1}$$, then a triangle with side lengths $${x^2-1}$$, $${2x}$$, and $${x^2+1}$$  is a right triangle.

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 1 > Topic A > Lesson 10Example 1

Algebra II > Module 1 > Topic A > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

A triangle has side lengths of $${(x^2-y^2 )}$$, $${(2xy)}$$, and $${(x^2+y^2 )}$$. Choose values for $$x$$ and $$y$$. Will the resultant three sides be a right triangle? How do you know?

 

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Include problems where students are asked to identify the value of $$x$$ and $$y$$ within a Pythagorean triples pattern. 

Target Task

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Problem 1

Generate three Pythagorean triples

Problem 2

Describe, algebraically, how you know that$${(x^2-y^2)}$$, $${(2xy)}$$, and $${(x^2+y^2)}$$ will always result in side lengths for a right triangle.