Polynomials & Place Value

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SWBAT write base-10 numbers in binary, hexadecimal, and other forms. SWBAT explain the analogy between the systems of polynomials and integers. SWBAT divide a polynomial by a binomial in simple cases.

Big Idea

Why do mathematicians confuse Halloween and Christmas? A good joke leads to some deep thinking.

What's Place Value?

10 minutes

A couple of nerdy jokes:

"There are 10 types of people in the world, those who understand binary and those who don't."

I'd like to have one of the students explain this to the rest of the class.  While the student is still at the board, I'd ask her to show us how to write the number twenty-five in binary. (Ans: 11001)  This example will provide a nice model for how place value depends on powers of the base.

"Why do mathematicians confuse Halloween and Christmas?" (Because OCT 31 = DEC 25!)

If the class needs a hint, I might write the dates for the two holidays, Oct 31 and Dec 25.  Now, what would a mathematician think when he sees "oct" or "dec"?  Eight and ten, of course!  31 in base-8 is equal to 25 in base-10!

An un-funny question:

"When is 123 equal to sixty-six?"

Be careful not to write, "When is 123 = 66?" since this is open to interpretation.  If the students aren't told that one of the numbers in written in base-10, they're going to have a tough time answering the question!

One way to solve it is by writing 123 as 1(x^2) + 2(x) + 3 and then setting the quadratic equal to 66. Students may also notice that the base for 123 must be more than 3 (since the digits 1, 2, and 3 are used) but less than 10 (since third digit is necessary).  This limits the possibilities to the integers from 4 to 9.

What's the point?

A key characteristic of the way we express the integers is place value.  There is a very close connection between place value structure and the structure of polynomial expressions.  In fact, you might say that when we write an integer, we're really writing a polynomial in shorthand (like when we do synthetic substitution).


The System of Polynomials

15 minutes
As the class continues, I use Socratic questioning to help my students understand the analogous relationship between the integers and polynomials.  I prefer this teaching method to lecturing because it helps students form a deeper conceptual understanding.  It takes some charisma to draw the students in at first, but it's easier to hold their attention because they're the ones doing most of the thinking.
I'll introduce the conversation with something like this:
"We've seen that the integers, in the way that we typically write them, are very much like polynomials.  In fact, you might say that our decimal notation expresses the integers as polynomials.  For instance, 235 is 2x^2 + 3x + 5 when x = 10.
"Now, we also know that the integers, along with the two operations addition & multiplication, for what is called a ring.  That is, the integers are closed under both addition & multiplication and multiplication is distributive over addition.  This tells us a bit about what the integers are by telling us how they behave."
Then, I'll begin my questioning:
"Do polynomials behave the same way, in general?  Do they form a ring?"
"The integers are closed under + & - and the identity element is 0.  Are the polynomials closed under these operations, and is there an identity element?"

     Yes, the identity is still 0, and this requires very little justification.

"The integers are closed under * and the identity element is 1.  Are the polynomials closed under *, and is there an identity element?"

     Yes, the identity is still 1, and this requires a brief justification using the distributive axiom.

"The integers are not closed under division.  Are the polynomials? No."

     Examine graph of quotient of two polynomials. It has a limited domain; no polynomial has a limited domain.

"So, can we safely say that the polynomials are like the integers?"

     Yes! Polynomials form a system analogous to the integers!

ANALOGY is one of the mind’s primary ways of understanding.  A new thing is seen to be like a familiar thing in significant ways.  ANALOGY is also a key to understanding mathematics.

Division of Polynomials

20 minutes

Another round of Socratic questioning to lead students to see that the set of polynomials is not closed under division.

"We've already covered a lot of ground in this lesson, but we have time to push ahead just a little bit farther.  So, let's talk about everyone's favorite: LONG DIVISION!  We've seen that the integers and the polynomials behave the same way under the operations of + and *.  Now what about division?" 

(At this point, I'll hand out the Long Division notes sheet and ask the class to take some careful notes - they'll be very handy later!  I'll also use my board so that the integer long division remains up for all to see as they work on the polynomial long division.)

  1. "How does division work for integers? Let's together divide 2275 by 7 using the long division algorithm."

I'll ask for a volunteer to walk us through the algorithm, and I'll be sure to ask the class to explain every step along the way.  It should be easy, right?

  1. "That takes care of the integers. Now, does division work the same way for polynomials? Let's try using the same algorithm to divide (x^3 + 7x^2 + 14x + 8) by (x + 4)."

I'll give the students about five minutes to try doing this on their own or with a small group to see if they can figure out how it works.

After about 5 minutes, I'll ask for volunteers to show the class how much progress they made.  It's likely that no one will have solved the problem completely, so I'll carefully walk the class through the process.  Following the same steps as with the integers - and pointing out explicitly that they are the same - we find that (x + 4) is a factor of the cubic. 

Finally, we can check our division by multiplying the two factors, or by evaluating the original cubic with x = -4 to see that it equals zero.  (This is related to the Factor Theorem and the Fundamental Theorem of Algebra!)

Practice Dividing Polynomials

10 minutes

Once all questions about the first example of polynomial long division have been answered, I'll had out Classwork for Polynomials & Place Value.  The students will spend the remainder (get it?) of the class period practicing the long division algorithm.

Each of these three problems involves a cubic polynomial divided by a linear binomial, and none of them leads to a remainder.  However, in the third problem the dividend is missing a quadratic term.  I won't mention this to the class because I want to see who notices it and what they do with it.  Call it a "teachable moment".

The students should work on these for homework, and we'll discuss the solutions at the beginning of the next lesson.