Proving Polynomial Identities

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Objective

SWBAT prove polynomial identities and use them to describe numerical relationships.

Big Idea

There are links between polynomials and geometry. Both branches of math use proof and a polynomial identity can be used to generate Pythagorean triples.

Warm-Up and Homework Check

20 minutes

As I circulate, checking homework with the homework rubric, I ask my students to compare their answers to the Polynomial Factoring Droodle, which was assigned for homework.  The warm-up that follows is a good application of MP3 because it asks students to make a mathematical argument.  I write the following prompt on the board and ask my students to work on independently when they have finished reviewing their homework.  

Work independently to disprove this statement with a counterexample.

Pick any two positive integers. Look at the sum of their squares, the difference of their squares, and twice the product of the two integers you chose. Those three numbers are the sides of a right triangle.

To be successful at this task, my students will have to both translate from words into mathematical symbols [MP2] and determine what makes a strong logical argument [MP3].

When they have worked independently for 10 minutes or so, I ask students to turn to a partner and explain their thinking.  The partners continue discussing and working together for another 3-4 minutes.  After this, I ask volunteers to come to the board to share any counter examples they have generated.  As a class, we work through the examples on the board, if any were offered.  We discuss whether the absence of a good counter example can be considered a proof.  

I then ask students to approach this problem algebraically.  Together we translate the statement into math symbols.  It says that

a2+b2, a2-b2, and 2ab are the sides of a right triangle.  

By the Pythagorean Theorem, [a2+b2]2=[a2-b2]2+[2ab]2

I ask my students to prove this identity by leaving one side alone and using algebra to make the two sides equal.  Although students will naturally want to perform operations on both sides I remind them that in proving identities we make our work more transparent by leaving one side alone [MP3].

 

 

 

Jigsaw-Style Student Exploration

60 minutes

For the next hour, students will work in groups to prove polynomial identities.  I start the activity by reviewing the proof from the warm-up with the presentation Polynomial Identities.  I remind my students that in proving identities, they are to use their algebra skills on just one side of the equation, working to make it match the other side [MP3].  

Although some of my students will have participated in a jigsaw activity before, it is the first time we are doing this type of activity together so I will take time to teach them how to participate.  First, they will work in small groups (color groups) to master one skill.  Next, they leave this group and become a member of a new group (number groups) in which they act as a teacher.  

In advance, I set up groups of 4 students that have similar algebra skills.  These groups will be known as the "color groups" because they will all be given a problem on the same color cardstock.  I also organize "number groups" that will be comprised of members of 4 different color groups that work well together.  Students know which number groups they are in by looking at the number printed on the back of their color card.  Each time I use a jigsaw activity, I need to set aside at least 30 minutes to work out the groups and so that I can allocate the Polynomial Identity Cards appropriately. 

The color cards contain the following polynomial identities

IVORY  a3 - b3 = (a - b)(a2 + ab + b2)          

GREEN  a4 - b4= (a - b)(a + b)(a2+ b2)         

BLUE  (a+b)3=a3+3a2b+3ab2+b3            

PINK a4+2(ab)2+b4=(a2+b2)2             

YELLOW  (2a+1)(a+1)=2(a+1)2-(a+1)      

ORANGE  (a+b+c)2=a2+b2+c2+2ab+2bc+2ac      

FUSCHIA a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ac)       

TO start the activity, each student is given a polynomial identity card and a copy of Activity- Proving Polynomial Identities.  The "color groups" come together and work together for 10 minutes to prove their identity. Each member of the group copies the proof onto their own copy of the activity

After all the color groups have finished their proof, I ask my students to rearrange themselves so that the number groups are together.  At this point, the color group experts present their proofs and answer the questions that arise in the number groups.  During this time, I circulate with a clipboard, taking note of who is struggling with the basic algebra skills required for the proofs.  This will help me know who to provide support to in subsequent lessons.  

After 30 minutes or so, students return to their everyday tables to complete the rest of the proofs.  I ask my students to display their color cards on their desk as they work so anyone needing support knows who they can ask for help.  

 

Whole-Class Discussion

15 minutes

After about 45 minutes of the jigsaw activity, I bring the whole group together for a discussion of the jigsaw exercise.  First, we discuss the mathematical content of the activity.  I ask for volunteers to summarize what it means to prove a polynomial identity.  I also ask for comments about which of the identities were the most challenging to prove.

Next, because this is the first jigsaw activity of the school year, I ask students how they felt about the activity.  I anticipate that some students will report that they like learning from their peers and others will say that they wish I would just write everything out on the board so they could copy it down!  There are always a range of opinions on this, which is why I am a strong advocate of switching things up from day to day.  

Exit Ticket and Homework Assignment

10 minutes

To check in on my students' understanding of the process of proving an identity, I distribute an Exit Ticket Polynomial Identities with a simple polynomial identity to prove.  I ask students to do this one on paper because typing math on the calculators can be very slow.  

For homework, I tell my students that they are getting a big break.  I am giving them a "throw back" to 4th grade homework.  Their assignment is Division and Factorization, which asks them to complete 6 long division problems and 4 factor trees.  I take a few minutes to make sure that they remember what these things mean.  I point out to them that if they have trouble remembering any of the steps they can certainly search the internet or ask their parents.  I will also make the solutions available on Edmodo so that they can check their work.

Depending on their reaction to the assignment, I may or may not tell them that they are practicing these things to get ready for our polynomial long division lesson tomorrow.