Fail Friday...on Saturday

I had been meaning to get help on this activity a while back, so Fail Friday seems to be the perfect opportunity for this.

Anyway...somehow I managed to totally suck at explaining intercepts this year in Algebra II.  Even after an entire semester with these kids, when I say, "What's the y-coordinate of an x-intercept?" all I get is *chirp, chirp, chirp.*

I wrote up this literacy strategy, that I was quite proud of.  I felt like they finally understood the algebraic definition of an x-intercept, and not just the geometric definition (i.e., I want more out of them than just "An x-intercept is where the graph crosses the x-axis.").  But...the next week I felt like we were back to square one.

Help!  How can I help them understand and generalize the concept of an intercept?  I especially want them to understand how factors and zeros are related.  What have you tried that you have had success with?  Class composition is juniors and seniors.

X-Intercepts and Factors by Rebecka Kermanshahi Peterson

Polynomial and Rational Function Review Day

Review days are a continual dilemma for me.  After each unit is finished and before each test day, I schedule a Review/Catch Up Day.  I'm not sold on this format, but I'm not opposed to it either.  Due to the fact that we live in Oklahoma, I definitely have to have a few built-in catch up days since we close our schools and colleges at the slightest sign of snow/"cold" weather.

I've been using these built-in catch up days for review, which seems natural enough.  The downside is that I often have very low attendance on these days. So, today I tried something new.  I offered something that nearly every one of my students will come to class for:  extra credit. 

You would think I would have had this revelation a while ago...alas.   

Math Mama wrote about a game of "Risk Your Beginning Algebra Skills."  I really liked the idea of the students having to wager a certain amount for each given question:  they have to be honest with themselves and decide how well they think they know something before seeing the answer.  So, I adapted Math Mama's idea and made a game of "Polynomial and Rational Function Jeopardy."  Before I showed them the answer slide, I would ask which students felt confident about their answer and then I had one of these students share with the class.  They were usually right on the money.  

I chose to do this in a slideshow format as opposed to a worksheet format because I know my students well enough to know that half of them would be working ahead and hence not contributing to the overall discussion (all my students are concurrent high school juniors and seniors).  Even if the overall discussion is not always where I would like it to be, some discussion still beats no discussion.

Overall, I think this review day was a success.  Students participated and hopefully got a good sense of what they needed to study for the upcoming test.

Here's the wager sheet I gave my students in Word format and PDF format.

Polynomial and Rational Jeopardy

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Discovering the Intermediate Value Theorem

Here are a couple slides that my students filled out that (I hope) helped them understand how to apply IVT to verify if a real zero exists between two given x-values.  Click slides to enlarge.

End Behavior Activities

We're currently studying polynomial functions in College Algebra.  Here are a couple activities I've done with the students to discuss end behavior.

Activity I:  The End Behavior Game
Not only do the students get to practice the Leading Term Test, but the teacher gets to enjoy a new variety of dance.

End behavior game

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Activity II:  The Polynomial Train
This is actually a twist on what I really did, but I think I will do it this way next time.

I'd start with a constant function (f(x)=1 in the example below) graphed using Desmo's calculator and ask a student to add a term to the function so that it would ________ to the left and ________ to the right.  The next student would be asked to add another term in order to change the function's tails to a new given end behavior.  After a few students, the function might look something like this:

So the first student was asked to add a term in order to change the end behavior so that it rises to the left and falls to the right.  The second student was then asked to add another term so that it rises to the left and to the right, etc.

I like using this calculator in class because the students can see how the graph changes as we change the output.

A final note...

One of my students showed me these hand motions and words (read from left to right) to go along with the end behaviors of polynomial functions.  He remembered it from years ago, so it must have stuck!  Maybe you've seen it before, but it was new to me.  I can just see a classroom of students taking a test and moving their arms in the air as they try to answer a question...

"Odd function"

"Even function"

"Nega..."

"...tively"

Two things about "Theorems about Zeros of Polynomial Functions"

I'm prepping for a lecture called "Theorems about Zeros of Polynomial Functions."  With the exception of the Fundamental Theorem of Algebra and a couple results from said theorem, the lecture should really be called "Stuff That Became Extinct with the Graphing Calculator but We Make You Learn It Anyway."

Two Observations:

First, our text states the Rational Zero Theorem as follows:

 Just out of curiosity...why not write it more like the following:

If f(x) is a polynomial with integer coefficients written in descending order and if p/q is a zero of f(x), then p is a factor of the constant term of f(x) and q is a factor of the leading coefficient.

Personally, I've stopped using the scary polynomial function notation all together in my College Algebra classes.

Second Observation:  Descartes' Rule of Signs.  There's no need for me to say anything, I'll just copy what I found on PurpleMath.com:

Descartes' Rule of Signs is a useful help for finding the zeroes of a polynomial, assuming that you don't have the graph to look at. This topic isn't so useful if you have access to a graphing calculator because, rather than having to do guess-n-check to find the zeroes (using the Rational Root Test, Descartes' Rule of Signs, synthetic division, and other tools), you can just look at the picture on the screen.

Bam.

This doesn't, however, change the fact that we still teach Descartes' Rule of Signs at my college.  Right now, I show my students the following: 

As you can see, with the exception of a few minor changes, I basically just copy what's in the text.  There's got to be a better way.  Please enlighten me.