Jacob Nazeck RIDGEVIEW CLASSICAL CHARTER SCHOOLS, FORT COLLINS, CO
Algebra II : Unit #4 - Higher-Degree Polynomials : Lesson #4

The Remainder Theorem, Day 2 of 2

Objective: SWBAT use the remainder theorem to identify roots of polynomial equations. SWBAT explain the relationship between factors and zeros of a polynomial equation.
Standards: HSA-APR.B.2 MP1 MP3 MP6 MP8
Subject(s): Math
60 minutes
1 Overview - 0 minutes

Having completed problems 1 - 3 of the handout, The Remainder Theorem, students are ready to formulate conjectures that will lead to a formal statement of the theorem.

A proof of the theorem is developed through Socratic questioning in a whole-class discussion, and then students use the theorem to answer a variety of questions about an unknown polynomial.

2 Formulating a Conjecture - 15 minutes

During the previous lesson (or at home) students should have completed problems 1 - 3 on The Remainder Theorem.  Now, it's time to begin formulating a conjecture about remainders and function values based on the results of problem #3.  This is problem #4 on the handout.

Using a Think-Pair-Share strategy, I will give the class about 2 minutes to formulate a conjecture on their own, then another 2 minutes to work with a partner to refine their conjecture, and then I will call on several students to share their conjecture.  For strategies for building a class discussion around student input, please see my Strategy Folder.

My first concern is that students identify the correct relationship - the remainder is equal to the value of the function - and my second is that they express it carefully.  The class will compare & debate various forms of the conjecture, I will encourage them to say precisely what they mean (MP 6), and together we will agree on a single wording.  Of course, I'll be quietly guiding them toward something very close to the formal expression of the Remainder Theorem that they will see in their textbook.

In the end, this conjecture will be written clearly on the board for all to students to copy into their notes.

3 Justifying the Conjecture - 15 minutes

Together, the class will do its best to explore the truth of the conjecture. (MP 3) I will open the conversation with something like this: "Wow.  This is really strange!  Why on earth would the remainder be the same as the function value?  I mean, we have four examples in which this is clearly true, but how can we claim that it's always true?"

Many students will recall from previous courses (and from common sense) that simply piling on more examples doesn't make a proof.  What we need is some kind of general justification, some kind of explanation of why this phenomenon occurs.  Depending on the class, this conversation can become very sophisticated.  To begin, it might be worth comparing the steps in the long division process with the steps in the evaluation process.  Next, you might consider taking a generic quadratic equation and dividing it by a generic binomial (see the resource).  Finally, you could walk your students through a more general proof like the second one included in the resource.  

Somewhere along the way, be sure to point out (or guide some student to point it out for you) that they've already proven a weaker version of this conjecture: the Factor Theorem, in which the remainder & function value were both zero.

Once the conjecture has been adequately justified, I'll erase the word "Conjecture" from the board and replace it with "The Remainder Theorem."  Everyone should copy the theorem into their notes.

It isn't Magic
Developing a Conceptual Understanding

At the beginning of this section, I put on a little show of being surprised by the pattern my students have recognized. This little act is easy to put on because I can recall very clearly just how surprised I was when I first encountered the Remainder Theorem.  I was being taught how to do polynomial long division, and the teacher simply said, "Oh, by the way, when you get a remainder you can put a point on the graph because the remainder is the value of the function at that point."  That was it, no explanation. It was as though the question why was completely irrelevant.  But to me, the question why was the most important.  Without knowing why it was true, this beautiful fact was just an unconnected and random bit of information.  It made all of mathematics seem arbitrary and unintelligible.

The message to me was clear: "Mathematics doesn't need to make sense.  Just learn the rules and come up with the answer."  Bleh.

In my teaching, I try very hard to explain everything.  When I can't explain something, I tell the class, "I'm not sure why that is, but I try to find out," and then I try to find out. When I don't have time to explain something, I'm honest with my students about that.  The message should always be the same: You don't really know something unless you know why it is.

The most important takeaway for my students is that facts in mathematics are always going to be true for a reason.  Math isn't arbitrary, and it isn't magic; it makes sense, and you can make sense of it.

4 Using the Remainder Theorem - 15 minutes

Students will work on problem #5 individually for 5 minutes, then in small groups for the remainder (Get it?) of the lesson.  The objective is to complete problems 1 through 5 and to be prepared to turn in the assignment at the beginning of class tomorrow.  Please see the video resource for more details!

An exit ticket might be an appropriate formative assessment for the end of the lesson.