# Use It

## Objective

SWBAT use polynomial identities to describe numerical relationships.

#### Big Idea

First you create it, then you use it...this lesson allows students to use the polynomial identities they just proved.

5 minutes

## Put it into Action

45 minutes

You will need copies of the Triple or Nothing Game Board, copies of the TRIPLE OR NOTHING rules and scorecards, plastic page protectors, markers and erasers for this section.  I've included a copy of a Completed Game Board and a Blank Game Board in its plastic sleeve for your review.

Game Time 25 minutes: As promised, I show my students the polynomial identity: (x^2+y^2)^2 = (x^2-y^2)^2 +(2xy)^2 and demonstrate that if they substitute two numbers for x and y, then simplify the equation until there are only three numbers left, waiting to be squared. (c)^2 = (a)^2 + (b)^2  Those three numbers, a, b and c, are a Pythagorean Triple!  Showing the final result using a, b, and c is intentional as a way of helping my students make the connection between the original identity and the Pythagorean more easily.  I walk through another example if necessary and then tell my students that they get to create their own triples by playing a game.  I explain that for the game they will be in groups of four, then distribute the game materials and tell them to review the instructions as a group.  I ask if there are any questions, then tell them they have about 20 minutes for their game. (MP1) While they're playing I walk around offering encouragement and assistance as needed.  The most common problem is students who argue about the rules and/or dispute the arithmetic of a teammate.  I settle the rules disputes and tell them to work out the arithmetic together! When I call time, I ask if anyone earned the bonus points, congratulate all the players and collect the materials.

Teamwork 10 minutes: I tell my students that now they get to move on to explore applications for other polynomial identities.  I explain that they will work with their back-partner to explore possible applications and/or other ways to interpret the remaining identities. (MP1, MP7) (for example what happens with (a-b)^2 or (-a+-b)^3?)  I play this up with the suggestion that finding a new pattern in one of these identities would be welcomed by mathematicians around the world! As they work, I walk around giving encouragement and redirecting as needed.  Generally working with a partner keeps things moving but occasionally a team gets stuck with where to begin.  This usually comes from a fear of doing the "wrong" thing, so I tell them that there are many ways to achieve appropriate answers. When there are only a few moments left or when most of my students have slowed to a crawl, I ask them to summarize what they've found in writing.  My hope is that they make some connections between similar identities, but even if they don't the process of exploring and writing about these identities builds a better understanding of polynomials in general.

## Wrap it Up

5 minutes

To close today's lesson I ask my students to write two problems that can be solved using a polynomial identity and show how to solve the problem. (MP2)  For example, a student might choose to write a problem about a square picture frame using the difference of squares to solve for a missing dimension, given the area of the center.   I explain that what I'm looking for is how they apply their chosen polynomial identities while working through to a solution. This gives them a chance to stretch their understanding of the identities a bit and also to demonstrate their skill at problem solving.  In order to write a good problem, a student needs to understand the process she/he will need to use to solve it.