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

View more PowerPoint from Rebecka Peterson

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"

How I Start Each Semester

I tell my students they "can do anything good...yeah, yeah, yeah."

Domain Restrictions of Rational Functions: And vs. Or

I've seen some of this recently:

I see the convenience of setting the denominator "not equal" to zero.  However, the math represented here is a little flawed.

What we know is the zero product property:

We also know that the inverse of this property holds:

In other words, when the product is not equal to zero, we need both factors to be different from zero.

So, when we teach domains of rational functions like the example above, we're getting a more restrictive domain than necessary.

That's my nitpicky rant for the day.

Around the World with Absolute Values

A few weeks ago I fell in love with the way Kate Nowak teaches solving absolute value equations and inequalities, as inspired by an article by Ellis and Bryson.

You should really read Kate's post because it's right on.  I found myself laughing out loud as she explained the common mistakes students make regarding these equations and her snazzy way of trying out a different method (different to me, at least).

The idea is to emphasize that absolute value measures distance.  I love this for two main reasons.

1. It places value (no pun intended) on the topological property of absolute value as the Euclidean metric on the set of real numbers.  If we see absolute value as just "making whatever's inside positive," we really lose the meaning of what absolute value can be used for in the topological sense.  "Making everything positive" is a shallow definition (and I'm sad to say I'm guilty of using it).
2. It eliminates the need for three cases.  You know what I mean.  "If it's equal to, set the inside equal to the positive and the negative; if it's less than, the inside goes between the negative and the positive; and if it's greatOR than, the inside is greater than the positive OR less than the negative."  Oy.  No matter how many times I'd justify those cases, and no matter how loudly I'd say "ORRR!" for the last case (shameful, I know), I would still get nonsense come test time.

So, today I got to teach absolute value equations and inequalities using this new strategy.  I was so excited.  And, much to my happiness, the lesson went very smoothly.  (Here are the lecture notes.)

To pound in the idea of what I've affectionately termed "Draw and Solve," we played a game the last twenty or so minutes of class.  It was a version of Around the World, which was seriously one of my favorite games as a schoolgirl (and I don't just mean inside the classroom).  Here's roughly how it went:  I would pull up a slide with a problem.  The first two students just had to draw the situation.  Whoever drew it correctly first became the defender and stayed at the board and the next student in line would become the challenger.  Now these two students had to solve the actual equation or inequality using the drawing the student before them had drawn.  The winner of this round would stay, a new student would come up and challenge, and we'd start again with a fresh slide.  So, the first slide looked something like this...

Thankfully, very few of my students are the "too cool for school" type, so they totally played along and seemed to enjoy themselves.  I'm sure giving extra test credit to the winners was helpful also.

The real metric (see what I did there?) will be how they do on their upcoming test.  I'm excited to see how it goes.

Happy Valentines Day!

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.

Quadratic Regressions with Angry Birds

Before we ended our discussion on quadratic functions, I wanted to introduce my students to quadratic regressions.  What better way to do this than modeling the trajectory of an angry bird?

This is the picture I projected on the screen.  There is more angry bird/quadratic function material out there on the world wide web that is better than this.  However, I couldn't find exactly what I needed with the materials I had (I wanted a screen shot of an angry bird, whose path wasn't yet finished, with a grid on top).  So, I made this one on Word using a picture I found on the internet.  Be nice.  It took much longer than one would think.

We chose some points that the top parabola passes through and then fit a quadratic regression using those points.  I then asked if the students thought the bird would hit the ice block (around (9.5, 4)).  The results for this varied depending on the class and what points they chose to input into their lists (which is fascinating).

Below gives a picture of the points we plotted in one of the classes, the quadratic regression, and the point (9.5,4), which was plotted post finding the regression.

There's more to be done here, I'm sure, but it's a fine start.

Parabolic Path of a Flying Q-Tip

This one I owe to one of my favorite teachers of all time.  It's a game we played in physics.  What you'll need for this is a drinking straw and 3-5 cotton swabs per student.

The lesson is on analyzing graphs of parabolas.  Since we just went over function transformations, we start with parabolas in the form f(x)=a(x-h)^2+k.  In this form, students seem to have very little trouble identifying the vertex.

So we graduate to f(x)=ax^2+bx+c.  Now, where's the vertex?  I used to do an elaborate presentation, completing the square for the general case and thus getting us back to the beautiful form above, and eloquently pointing out what h and k now are.  And then I came upon this beauty:

"Where is the x-coordinate of the vertex in relation to the two zeros?"  I ask.

From here, we have a little discussion, and eventually decide to average the two zeros to come up with the x-coordinate for the vertex, which turns out to be -b/2a (horrah!).

Now we practice.  I give the students a few quadratic functions and ask them to tell me all sorts of things about its graph.  The point I try to emphasize is that once you know the vertex and the direction the the parabola opens, you know a whole host of other information (such as the axis of symmetry, the min/max value and where it occurs, where the function is increasing/decreasing, and its range).

Now for the game.  I tell the students to take out the straws and Q-tips I passed out in the beginning of class.  I play this video and tell them whoever can shoot the cat with a Q-tip gets extra credit:

I gave out three Q-tips per students, but will most likely up that to four next semester, depending on how many students I have.  The students love this part of the lesson--the athletes in the room seem to take special pride in hitting the Nyan Cat.

And now back to the math.  We talk about how the path of their Q-tip takes the form a parabola.  We discuss if a should be positive or negative.  I then give them a model such as f(x)=-4x^2+4x+25 and explain that the path of their Q-tip could be modeled by such a function, where x is the time after the Q-tip's launch in seconds and f(x) is the Q-tip's height in inches.  (No, the model isn't perfect, but the numbers are fairly easy to work with and it gets the point across.)  We play with the model a bit, finding the ever-important vertex, among other things.

In the past, I've had students shoot at me during the lecture.  They love this.  Until they have to do the homework that night and realize they caught zilch during the lesson.  So, I tried the cat video this time.  It's probably not quite as entertaining, but in the long run, I think it may work better.

Transformation Scavenger Hunt

Last week was the wonderful function transformation lesson.  I say wonderful with a bit of sarcasm as this section tends to overwhelm me a little every semester.

Let me tell you about symmetries.  Let me tell you about translations.  Let me tell you about reflections.  Let me tell you about stretching and shrinking.

All in under an hour.

This semester, the lesson did go a bit better, I have to say.  For one, the students seemed totally into transforming graphs of functions to make them look completely different than what they started out as.  Kudos to them.  For two, I adapted Kate Nowak's buried treasure idea to fit into our lesson.

I took the twenty-five desks in my classroom and handed out a worksheet that had this on the top:

So each desk corresponded to a point on the Cartesian plane.  I gave my students a set of points as well as a function transformation.  They were to locate the points and transform them correctly.  I told them the treasure was at the point furthest to the left, for example.  Underneath the desk, I had taped an index card that said something like, "Congratulations!  Hint 1 out of 3 correctly found."  Once they found all three cards I rewarded them by playing a favorite YouTube video of mine (which, I admit, was not math-related).

All in all, the scavenger hunt was a success!  This semester, I had the class split into groups.  Once a group felt like they had the correct point, I had them tell me their answer.  If they were wrong, I told them to try again; if they were right, I had them wait for the rest of the class to get to the right answer.  This of course forced the faster students to wait a little, but that was my only complaint about the activity.

I plan to use this again next semester when we come across function transformations again, as well as incorporate Desmo's graphing calculator, which has (wait for it) SLIDERS.