A case I hadn't given much thought to until recently

A recent run-in with an old AP Calculus question has got me thinking about relative extrema lately, specifically in the case of a removable discontinuity, as in this graph:



Question:  If the graph above represents a function f, does f attain a relative minimum at x=b?

Noticing and Wondering with the Binomial Theorem

This is my first year teaching Pre-Calc.  However, with the exception of our trig unit (which, granted, is a good portion of the class), I've taught most topics we cover in Pre-Calc.  But, today's lesson was on the Binomial Theorem, which I had never taught before.  As I was reading up on it, I found myself noticing and wondering.  There's so much to explore.  At first glance, do a bunch of expansions look all that thrilling?  Maybe not.  But, the more you dig into it, the more patterns you begin to find.  So, I decided to put my students to the challenge, too.  This was their warm up today:

I gave them 3-5 minutes.  And then I started calling on people to share, writing their thoughts on the board so everyone could see.  They were hesitant at first but grew more confident as we went on.  After I had called on several kids, I asked if anyone else had something s/he wanted to contribute.  These are the lists we made in my two classes:

Mostly, I just wanted to share my students' thoughts, because I thought they did a great job for this first-ever notice/wonder assignment.  Also...the second class's "wonder" was, of course, the very nature of the lesson, so...mwah!

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