## Reflection: Developing a Conceptual Understanding The Remainder Theorem, Day 2 of 2 - Section 3: Justifying the Conjecture

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.

It isn't Magic
Developing a Conceptual Understanding: It isn't Magic

# The Remainder Theorem, Day 2 of 2

Unit 4: Higher-Degree Polynomials
Lesson 4 of 8

## Big Idea: By observing regular patterns in their work, students formulate conjectures leading to the Remainder Theorem for polynomials.

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Standards:
Subject(s):
Math, Polynomial and Rational Functions, Algebra 2, master teacher project, Polynomial Long Division, Remainder Theorem, Polynomial, Polynomial Operations and Functions
45 minutes

### Jacob Nazeck

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