This is the first of two lessons focusing on polynomial identities. The purpose of this lesson is for students to investigate expressions in order to understand that a polynomial identity is defined as an equation where the expression on the left has the same value for all possible inputs \(x\) as the expression on the right.
In the warmup and following activity, students explore the difference of the squares of two consecutive integers, calling back to the identity \(a^2  b^2 = (a+b)(ab)\) they have worked with in earlier grades. Then students explore several cases of an identity where application of the distribution property leads to an expression with fewer terms than might be expected (MP8). This particular identity will be revisited in a future lesson where students derive the formula for the sum of a geometric series.
Lesson overview
 23.1 Warmup: Let’s Find Some Differences (5 minutes)

23.2 Activity: A Closer Look at Differences (15 minutes)
 Includes "Are you Ready for More?" extension problem
 23.3 Activity: That Expression is How Big? (15 minutes)
 Lesson Synthesis
 23.4 Cooldown: Is This an Identity? (5 minutes)
Learning goals:
 Comprehend what a polynomial identity is.
 Prove some common identities.
Learning goals (student facing):
 Let’s learn about polynomial identities.
Learning targets (student facing):
 I understand what an identity is in mathematics.
Glossary:
 identity  An equation which is true for all values of the variables in it.
 Access the complete Algebra 2 Course glossary.
Standards:
 This lesson builds towards the standards: CCSS.HSAAPR.C.4CCSS.HSASSE.A.2MS.ASSE.2CCSS.HSASSE.B.4MS.A.SSE.4MS.AAPR.4
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.