Sunday, September 28, 2014

Formal Definition of the Derivative

Howdy, fellow AP Calc teachers!  You know those derivative questions on the AP exam that are disguised as limits?  Here's a worksheet to help students be able to recognize these questions and hopefully gain some fluency in moving from one form to the other.

I gave this before the kids learned any shortcut rules.  After they learned the rules, they filled in the answer column.

Sunday, September 21, 2014

Local Linearity

I haven't posted at all since school began over a month ago.  It frustrates me that the periods I do some really great work in the classroom (or at least, I think so) are also typically the periods I have zero free time and hence the best blog posts potentially go unwritten.

Le sigh.

Well, regardless, here's something we did in calc a couple weeks ago...

A big idea I want kids to come out of calc with is that if we zoom in on a differentiable function long enough then the function, no matter how "squiggly," will start to look like a line.  Why do we care?  Because lines are crazy easy to manipulate and make predictions from.  The key is, if we get too far away from our "zoom point," the line we came up with will no longer be useful to us.

To help kids grapple with this idea, I split them up into groups and gave them a function and an x-value.  They were to graph the function on their calculators and then zoom in on the given x-value until they felt the function looked like a line.  At that point, I asked them to find a few things:

  • The two end points of the line segment shown in their viewing window (using the TRACE feature)
  • The slope of the line using these two points
  • The local linearization (i.e., the equation of this line)
They were asked to do this for four different x-values.  Here are the slips of paper I cut up and gave each group:

As they turned in their finished slips, I started to type in their lines (with restricted domains) into Desmos.  I couldn't type as fast as they were finishing, but this is one of the pictures we started to produce.

Hey guys...what function do you think these two groups had?  y=x^2?  You bet!  Could I have used that first red line for the whole function?  No?  Why not?

I think it really made sense to the kids.  In the future (NOW THAT DESMOS HAS A FREE APP!!!!!!!!!!!!), I think I will share a link with each group that has the same function and then THEY can type in their lines.  I'll have to show them how to restrict domains so I'll probably only have each group come up with two linearizations as opposed to four.

Here's the homework I made that went along with it.  Not sure the homework is much to write home about, but at the very least it gets the kids working with tables, which is a representation they need to be more comfortable with:

Tuesday, August 5, 2014

My Favorites {TMC14}

I hope everyone who did a My Favorites talk at TMC14 has blogged or will blog about it at some point.  I shared two favorites and wanted to document them here, for those who had any questions or for those who didn't get to attend this year...

#1:  Friday Letters

I stole this entirely from a middle school teacher (now assistant principal) in my district, Scott.  Every Friday, my kids have the choice to either do the warm-up on the board or write me a "Friday Letter" (a letter to me from them about anything they want to talk about).  In the beginning of the year, I remind them of this choice often.  When they enter the classroom, the board might look something like this:

At first, this is a pretty novel concept for most of the kids.  Plus, they get to get out of the warm-up, so most everyone writes a letter.  As the year progresses, I remind them less and less (and consequently fewer kids write).  However, I keep some promises:

  1. The mailbox will always be in the back of the room for you to put letters in.
  2. I will check the mailbox every Friday.
  3. I will personally respond to every letter I receive.

Yes, #3 can be daunting at times.  At first, I got a lot of letters that were mostly just, "Hey Mrs. P!  Hope to see you at the football game tonight!" or "What's your favorite Harry Potter book?"  And I had promised to respond to every letter, so I did.  But, as the year progressed, I received fewer of those kinds, and I mostly only got letters from kids who really enjoyed communicating through writing.  I would have kids write their letters at home on Thursday night so that they could put them in the mailbox on Friday (several letters were a full page, front-and-back).

I got funny letters and heart-breaking letters. Sincere letters and goofy letters.  But each letter gave me insight into a kid; insight that I wouldn't have gotten any other way.  I kept each letter in a big, green binder that I'll add to this year.

I found that my quietest (usually very successful) kids would have the most to say.  When I would check in with them in class and ask how things were going, they'd reply with a quick "Oh, I'm fine! No questions yet!"  But then I would get these novels from them in their Friday Letters.  It was a way for me to connect with kids that I really don't think I would have connected with otherwise, or--at least--not on that level.

Some people asked me how I would respond to the letters.  You know those yellow legal pads of paper?  You can buy them in a smaller size (5"x8"), and that's what I used to write return letters.  I tried to fill at least the front side half-way.  Sometimes, though, I would fill both front and back fully...depending on how much the student had to say and how much I had to say in return.

#2:  Mathematician Spotlight

I've actually blogged about this before here and here.  This is a way I incorporate a little (emphasis on little) history and language arts into my PreCalc classes (though I think this could be done at just about any level for middle school, high school, and college students).  Essentially, my kids research a mathematician for some extra credit for each unit test.  Last year I also gave them a quote by the mathematician and had them defend or dispute the quote.  I think this year I will have them find a quote on their own, instead of giving one to them.

I made a new sign for this year (above).  Feel free to print and use it if you'd like (click here to download).

Tuesday, July 29, 2014

Pretty Please Join Us

I had an amazing time at TMC this past week.  I hope to write more about it soon.  I'm not quite there yet, though...

However, as a result of TMC, Levi Patrick (@_levi_), Oklahoma's Director of Secondary Mathematics, asked some of us Oklahoma bloggers to talk about why we love the MTBoS, in hopes that other Oklahoma teachers would jump on this bandwagon.  So here we go:

And for those who prefer to read, the transcript (more or less):

Hi!  My name is Rebecka Peterson.  I teach algebra through calculus at Union High School in Tulsa, and I want to take two minutes to tell you about an amazing group of math teachers who have changed the way I teach.  They call themselves the “MTBoS,” the Math twitter blogosphere.  We’re a group of math teachers who interact online (mostly through blogs and Twitter) to help each other grow in our respective classrooms.  It’s a virtual PLC.  Everyone’s story is a bit different, but here’s how I got started:

I started reading blogs about two years into teaching.  I think it all started one day when I dangerously Googled the words, “How to teach absolute value equations,” and stumbled upon Kate Nowak’s blog, A Function of Time.  I was totally captivated by the way she taught these equations and immediately started reading more articles—both by her and by other bloggers.  And I was hooked. 

I lurked for a few more months: at first I was solely a reader.  Then, I got brave enough to add a comment here or there.  As I continued to read, I was simultaneously impressed and overwhelmed by these amazing teachers.  They were so good at their craft.  These bloggers became my heroes.  So, much like a little sibling, I decided the best way to become like them was to start my own blog, too. 

I started blogging early in 2012.  At the time I was teaching at the college level, but most of the teachers I interacted with online were high school teachers.  To be completely honest, they were a really big part of my decision to accept a high school teaching position.  They were so passionate and so encouraging and so willing to share that I felt like experiencing what they experienced day in and day out would surely only lead to further growth.

And it did.  I just finished my fifth year of teaching, my second at the high school level, and I wouldn’t want to be doing anything else.  While I have really amazing coworkers, together, we still only make up a very very small piece of the pie.  So, I love interacting with other teachers online because you have that many more people investing in you and wanting to see you grow.

One of the most rewarding things about blogging is once I publish a post, others will take an idea I wrote about and make it so much better, or tweak it so that it fits their classroom needs.  In so doing, it’s possible that you can positively affect other teachers or students far outside your own school.  And being math teachers, I think we can all appreciate the ripple effect that can take place.

In closing, I just want to encourage you—you don’t have to jump in with both feet right away.  You don’t have to blog AND use Twitter.  Most of us start by just reading.  Read posts that you find interesting and that you think will be beneficial to YOUR classes.  And, when and if you’re ready to participate more, just create a blog or Twitter account and see where it’ll take you.

Welcome to the MTBoS!  I hope you’ll grace us with your expertise and questions.


Check out @mathequalslove and @druinok's videos, too!

Thursday, June 26, 2014

Slope Field Activity

I’m getting ready for Twitter Math Camp's calculus working group (ah!).  We’ve been asked to bring some activities to share with the group, so I’ve been frantically searching my blog this afternoon for stuff I can contribute.  I was typing up a list when I realized I don’t have a whole lot for the second semester of Calc AB.  I imagine a lot of that has to do with the fact that we’re in review mode for half the semester, but still.

However, I did realize that there was one pretty good lesson on slope fields that I didn’t blog about.  I probably didn’t write about it because I stole it 100% from my APSI instructor last summer.  Nevertheless, I’m not feeling that trepidation currently.  This magic should be shared.

  • On the board, draw or project a blank Cartesian plane along with a differential equation.  There should be at least as many integer coordinates as there are students:

  • Give each student a card with a coordinate on it.  {If you're as Type-A as I am, here are cards you can print for up to 35 students.  And if you're REALLY Type-A, you can print them on card stock, laminate them, and cut on the solid lines.  I love laminated cards.  Laminated cards make me feel like I just insured a valuable asset. Moving right along...}
  • Each kid figures out the slope of the tangent line at the given point and draws a tiny line segment with that slope at the given point.
  • When everyone is finished, they’ve all contributed to a graph that looks something like this:

My class's actual slope field;
not perfect, but whose slope field is?

For this example, we discussed questions like:
  • What's the pattern for the slope to be zero? Why?
  • What is the slope doing to the left of y=x?  Why?
  • What about to the right? Why?
And I mean that was pretty much all they needed in the way of instruction.  Not that it's all that complicated to begin with, but this was a nice, everyone-get-up-and-contribute type of lesson.

Again, I can take zero credit for this.  But I thought it was worth sharing.