My Unit on Rational Functions (Algebra II)

Disclaimer--this unit is fast and very calculator-heavy.  It would need a good deal of reconstruction for an Advanced Algebra II course.  Nevertheless...

Part I:

Review of asymptotes via Asymptote Bingo

Part II:

Introduction to rational functions via this foldable:

Rational Functions Foldable by Rebecka Kermanshahi Peterson

*I think you could use this in an Interactive Notebook if you just deleted Example 3.

Part III:

Exploring rational functions via Desmos

This was my favorite.  Oh, Desmos, how I love thee.  I wrote this literacy/technology activity for my students and then we headed to the Math Lab together to work on the computers:

Rational Functions With Desmos

We have really nice, big screens in the Math Lab so the kids were able to get beautiful and clear pictures of these functions, which (I think) a typical handheld graphing calculator can't quite provide.  Here's what I loved:  The kids would graph the function in question, for example this:

And then they were asked to analyze.  I asked them to graph all their asymptotes and highlight all intercepts.  So, if they accidentally said that the horizontal asymptote was x=0, when they graphed their answer, they (usually) immediately identified their mistake and made the appropriate corrections.  (Or, at the very least, they raised their hands and told me, "This doesn't look right to me...")  If done correctly, their ending picture should have looked something like:

So beautiful and clean!

Part IV:
Solving rational equations through graphing and technology

Including review and assessment, I spent just over a week on this unit (like I said, it was fast).  But I'm pretty happy with it--especially our day in the Math Lab.  I worked out some issues with the activity, so I'm interested to use it again (I want to try it in PreCalculus) and see how it goes the second time around.

Asymptote Bingo

We're starting rational functions next week in Algebra II, but I knew my kids weren't super strong with asymptotes yet.  Truth be told, we still have a hard time recognizing that vertical lines will be an x=____; horizontal will be a y=____.

So, we played some Bingo on Friday, and I'm hoping for a solid start on rational functions tomorrow.

We play Bingo a good deal in Algebra II, so the kids are pretty familiar with my set-up by now.  Here's how we do it:

1. On a personal whiteboard, draw four horizontal lines and four vertical lines, creating a 5x5 game board.  Mark the middle box as your FREE SPACE!
2. Fill in  your board with these equations.  You may fill them in however you'd like, but you must use all 24 boxes--so keep track as you go!

Now we're ready to play!

I showed a graph of a function and then told the kids to cross out the correct asymptote.  The first several functions were strictly exponential and log functions--which we've already studied.  Anytime someone got a Bingo, s/he got a piece of candy (thank you, dear parents and guardians).  We played all hour, which gave nearly every kid a chance to get at least one piece of candy.  I kept track of the equations we had used and had the students call off their equations so I could check their answers.

After awhile the students started asking, "When are we going to get to the ones with two or three equations?  I need one for a Bingo!"

Mwahahahahaaaaa!  They were asking to learn about rational functions, and they didn't even know.

Before they knew it, they were analyzing the asymptotes of rational functions without any real instruction from me.  Just good progressions from what they already knew to what they needed to learn.

And now I feel at least a little better about their knowledge of asymptotes.

If you want the Notebook Bingo file, click here.

A good little Pre-Calc Problem

Write the equation of a rational function with the following properties:

• A zero at x=4
• A vertical asymptote of x=2
• A slant asymptote of y=x-3

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

View more PowerPoint from Rebecka Peterson

Exploring Rational Functions

Yet another idea I've stolen completely from Kate Nowak.  The unit we're currently working on in College Algebra is on polynomial and rational functions.  I get one day to cover rational functions.

One day to cover possibly the most fun word in all of algebra:  asymptotes.

Let's face it--the day our math teachers condoned the word ass was a good day for all of us.

Anyway, here are the lecture notes we discussed together in class.  If you haven't read Kate's post, read it first.  I didn't change much.  I just added a few things that we need to cover for our college-wide cumulative final, such as horizontal and oblique asymptotes.

I was very rebellious with this section and spent an entire day and a half on it.

4.5: Rational Functions