Polynomials

Lesson 12

Math

Unit 3

11th Grade

Lesson 12 of 14

Objective


Use polynomial identities to determine Pythagorean triples.

Common Core Standards


Core Standards

  • 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

  • 8.G.B.7

Criteria for Success


  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


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

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Anchor Problems


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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Target Task


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.

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or 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. 
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Lesson 11

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Lesson 13

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Polynomial Features and Graphs

Topic B: Operations with Polynomials

Topic C: Polynomial Extensions

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