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.

Introduction to Tangent Lines

I've been loving this introduction to calculus that we're doing with our Pre-Calc classes currently.  I don't know about you, but when I was in Pre-Calc, I didn't do any calculus.  Not a single thing.  I had no clue what a limit was, and certainly not a derivative. My Pre-Calc class was pretty much just trig, trig, and more trig, with a bit of "advanced" algebra thrown into the mix.  (I'm not complaining though--it was a great class, honestly...and I'm told I should be thankful that I'm young enough to even have had a class termed "Pre-Calculus.")

Anyway.  All this to say--it's darn exciting introducing kids to concepts such as limits, derivatives, and integrals because they're so powerful and beautiful...and so unlike other stuff we teach (no?).

So, a few things I'd like to share from this week.  Nothing's super original, but I did put a lot of time and energy into making them work for my students.

First:  Visualizing secant lines turning into the tangent line via Desmos.  Again, I know there are plenty of applets out there, but I couldn't find any that my students in the back of the room would be able to see.  Also, I wanted to input my own functions.  Also, I wanted to create it because it's fun and allows me to use mathematics.  So, here you go.  Slide a, change the function, change the point of interest.  Best of all, put it in projector mode so everyone can see--even the kids in the back.

Second:  We had an extra day built-in for tangent lines, so during collaboration, I asked if we could create a packet that introduces the kids to how to draw those lines exactly.  And how does the algebra relate to the geometry?  My department head and I discussed the objectives, and then she miraculously turned our words into this beauty:

Tangent Line to a Curve

Third:  This Warm Up that I rather like (Day 3 of Tangents):

Fourth:  I used these sites so the kids could get some practice visualizing what the derivative function would look like without taking the time to actually find it algebraically.  I love exercises like this because they truly require deeper thinking.  You can't bs your way through them.

• Some multiple choice practice
• Derivative plotter

Solving Exponential and Log Equations Flow Chart

I guess I've been kinda into flow charts this year; I created another one for solving exponential and log equations.  The idea is that kids start with the top box, if they can't complete that task, then they go on to the next box.  We put examples in each box.  I used this for both Algebra II and Pre-Calc this year.

It's not flawless, because mathematics requires more creativity than a flow chart can provide.  But it gets the basic ideas across.

Solving Exponential and Log Equations Flow Chart

Newton's Law of Cooling :: A Murder Mystery

You know those lessons/projects you give your students that you look back on and you're like, "Wow.  That wasn't half bad"?  I had one of those recently in Pre-Calculus.

We've all seen those exercises in textbooks where students are supposed to figure out the time of a person's death using Newton's Law of Cooling and given certain temperatures and times.  I always liked those problems, but never really knew what to do with them, more than just present them and say, "See!  Math IS applicable to real life."

Then a colleague of mine showed me a literacy activity adapted from Key Curriculum Press.  I found a version online that I used (but it was a direct link to the Word document, so I don't know to whom to give credit!).  The first page is what I found online (I added Newton's Law of Cooling to the bottom); the second page is the instructions for the kiddos, which includes the rubric:

Sherlock Holmes Mystery by Rebecka Kermanshahi Peterson

To start out, we first had to watch a trailer for BCC's Sherlock (LOVE):

I gave the students about a half a day to figure out the math and solve the murder.  The next day, I loaned out a laptop cart from the school and the kids finished the story, working in groups of 2-3.  I had the students submit their posts on a blog I created for our class via kidblog.org.  My principal told me about kidblog, a class-friendly version of Wordpress, and I absolutely love everything about it (except its name).  Students don't have to register or sign in with an email account:  you just set up usernames and passswords (which can be done in a jiffy) and then they can log in.

On the blog, I posted a sample writing that I found here.  (The math is a little off, so be sure to fix it if you use this link--the final t should be negative.)  This really eliminated the "I don't get what you want us to do!" comments because the students had an example with which to model their writing.  In fact, I didn't get a single such comment (kuddos, kids).  However, I also protected this sample with an extra password:  students could not get into the post until they had solved the crime, as the password to the post was the time of death (see, kidblog is awesome).

Once all the posts were in, I gave the students a couple days to go back in and comment on their favorite posts.  The posts with the most comments received some bonus points.

I was honestly blown away by my students' response to this assignment.  Their stories were original, entertaining, and included the required mathematics.

There's obviously room for growth here on my part, but for the first go-around, I was incredibly pleased with this activity.  Next time, I may make the crime a bit harder to solve, and I may give different versions.  We'll see how motivated I am.

Out of respect to my students, I don't want to post the password to the blog here.  But, if you'd like to check out their stories or the blog for instructional purposes, feel free to tweet me (@RebeckaMozdeh) or email me (rebecka dot peterson at gmail dot com).