You may want to make a section of heavy string with 11 evenly spaced knots and tied at the site of the 12th knot for this part of the lesson or you can just sketch it. I start this lesson by asking if anyone knows how the Egyptians made sure the angles in their Pyramids were 90 degrees. If a student already knows about the knotted string, I take my string out and demonstrate how it works. If not, I explain, using my string, that a triangle with sides of lenghts 3, 4, and 5 always makes a right triangle. There are usually a few students who challenge this, so I ask them how they check to see if a triangle is a right triangle. When they respond with the Pythagorean Theorem, I suggest that they try it for my 3-4-5 triangle and see for themselves that it works. (MP1, MP4) The story of the Egyptians gives a good example of a use for Pythagorean triples that most of my students can relate to, which I explain further in my Use It video. To reinforce that connection I ask if anyone has every had to create a right angle for anything and get responses ranging from marking out a fenceline to helping cut linoleum to marking a baseball field. I then ask if anyone knows the special name for the sides of a right triangle that are all integers. Again, if anyone knows the answer I congratulate them and otherwise I give the name and ask my students if they know any more Pythagorean triples? This results in some of my students trying to find more, so I ask if they would like to know how to generate more triples other than by trial-and-error. By now, if I've done this right, most of my students are ready to learn this new method.
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.
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.