Adaption of AP Calculus Questions

This is my first year teaching calculus through the AP curriculum, and I love, love, love, LOVE it.  It's such a great mix of pure and applied math; I really feel like it supports teachers as we try to cultivate thinkers in our classrooms...and not just regurgitators.

In PreCalculus, we are now starting to discuss actual calculus topics:  limits, formal definition of the derivative, and approximation methods for area under a curve.  I was reading this article from the College Board about vertical alignment, and it got me thinking about how we could be exposing our Algebra 2 and PreCalculus kids to past AP Calculus questions.

And so I created these questions, which are adaptions from the 2012 and 2011B AP Calculus exams (both questions appeared on the AB and BC exams).

AP Area Questions by Rebecka Kermanshahi Peterson

Does anyone else have previous AP questions they've modified to fit earlier classes--not just for calculus but possibly also for statistics?  Or, do you have "vertical alignment" in your pre-AP math courses?

Some {Minor} Improvements on the Teaching of Limits

We just got done introducing limits in precalc.  It's a little more fun teaching it in precalculus as opposed to calculus since limits are review for my calc kids.  In precalc, my colleagues and I take a week to introduce this topic as we use a three-fold approach: understanding limits graphically, numerically, and analytically (and I would throw in verbally also).  To me, this is a fundamental concept in modern mathematics--to be able to discuss values that are either (1) tending towards infinity or (2) getting infinitesimally close to another value.

Infinity is the savior of calculus, and limits are the heartbeat of infinity.  And so, while my kids learn about limits more from an intuitive approach rather than a rigorous epsilon-delta approach, I still feel like they're doing good, valuable mathematics.  I know other calc teachers disagree with me on this one, but that's my stance.

So a few things I've done this time around that were successful (though none are my original ideas by any stretch of the imagination):

#1:  Creating Graphs
On Day 2, I had kids create all kinds graphs that satisfied certain criteria.  For example, "Sketch a graph such that the limit as x approaches 2 of f(x) is 4 but f(2) doesn't exist."  Or, "Draw a graph such that the limit as x approaches 3 from the left of f(x) is 1; the same limit from the right is -1; and f(3)=5."  The kids drew on personal whiteboards, and when I saw one I liked, I would ask the kid to put it on the Smart Board (or ask for volunteers).  Each time we had at least two examples on the board for the whole class to analyze, and I encouraged the kids who couldn't quite come up with the graph on their own to now try to make one or even replicate one that was shared by a classmate.  The good thing about having at least two graphs to look at on the Smart Board is that you can ask, "What things are the same about these graphs?  What things are different?  For the things that are the same--did your classmates HAVE to draw their graphs like that or could I change that aspect and still satisfy the given criteria?"  Really good conversations came from these graphs.  I started out pretty basic and gradually gave them harder ones.  By the end, I think every kid was able to create graphs with the given the criteria, which is exciting because my students have been somewhat unsuccessful at this in years past (because I haven't made it a big part of the learning process, which is a shame).  The last graph I had them draw was something like, "Sketch a graph such that the limit of f(x) as x approaches -2 from the right is 1; the limit as x approaches -2 from the left is 1; the limit as x approaches -2 is dne."  Of course, this is an impossible task, but it was highly amusing watching their faces as they read the question with bewilderment.  They inevitable tried to draw it, but there was a lot of erasing going on. ;)  I made sure to let them be the first ones to say something about the difficulty of the task.  Which brings me to Thing 2...

#2:  Talking about Limits

I recently attended a seminar on discourse in the mathematics classroom.  Getting kids to talk about math is something I'm passionate about and something I'm trying to get better at, so this was right down my alley, and I was able to absorb some really good information.[1]  From this seminar, there are two practices I'm trying to implement consistently:

1. Don't show approval for a correct answer nor disapproval for an incorrect answer right away.  Instead, have the kid who gave the answer explain his/her reasoning regardless of whether or not the answer is correct.
2. Don't let a kid opt out.  If all else fails, at least have the kid repeat the correct explanation of another student.

Limits turned out to be a perfect platform for me to practice both of these skills.  One of the hardest hurdles to overcome for my kids seems to be navigating removable discontinuities.  They see a hole at x=a and assume that because f(a) doesn't exist, the limit there doesn't exist either.  I can say, "a closed or open circle doesn't affect the limit" until I'm blue in the face, but that doesn't do the trick for all kids.  So, I tried to throw in a lot of practice with this concept and when a kid would say that the limit dne, I would have him explain his reasoning.  Without fail, his classmates would correct him (kindly, I might add) and boy howdy, it's so much more fun hearing explanations come out of their mouths than my own.

In one instance, I had a student that had been absent and when called upon, he was having a hard time getting to the right answer and an even harder time explaining his logic.  We soon looked at another example with a similar problem (removable discontinuity); this time he could get the right answer but still couldn't explain (but was starting to get the hint that I wasn't going to let him off the hook).  So, I had his partner explain and then immediately asked the original student to explain.  Everyone laughed as this was the third time I had asked the same question from the same kid in about a 2-minute time span, but the kid repeated what his partner said and vowed that he, along with his whole class, will now certainly answer correctly on the next test.

While that's a very simple situation and while it seems easy to implement this kind of discourse, it really isn't for me.  Yes, I love getting my kids to talk about math, but I find it takes extreme intentionality, perseverance, and patience on my part.

#3:  Limits Algebraically--Four Scenarios

The last way we learn to evaluate limits is analytically.  I start by telling the kids that we always want to begin by plugging in what x is approaching into the given expression,[2] because, ideally, our function is continuous there.  And if this is the case, we're happy and we can move on to solve the world's next problem, which surely involves limits.  This is what I ask them to write in their notes (which I'm pretty sure I learned from someone at AP Summer Institute):

[3]

My kids totally ate up the 0/0 becomes "do more work."  I mean, like literal gasps were heard in every class.  I know this is a little trick-sy, which I don't love, but I feel that once we've talked through each scenario, the kids have a fairly good grasp on the why.  Furthermore, they get a pretty firm handle on the fact that 0/0 is an indeterminant form, so they can't just assume that the limit doesn't exist...they must do more work to find out the true value of the limit.

#4:  Graph Pictionary
After we talk about limits at infinity (next week), I plan to use this activity from the Study of Change blog.  Kids get into groups of two:  one person is the "Board Partner" and the other the "Drawing Partner."  The Board Partner looks at the graph I show on the board and describes the graph using words only (hands must be folded on desk!) while the Drawing Partner, who is facing the back of the classroom, draws the graph to the best of his/her ability.  And then, we switch roles for the next graph.  The hope is that kids utilize correct mathematical vocabulary, as this will be one of the most helpful strategies to get graphs looking right.

Here are the graphs I'm using...hopefully I can hear the word limit a lot a lot a lot.

Graph Pictionary

And those are my thoughts on the teaching and learning of limits.  What are yours?  Do you have any kinds of problems that get your students talking and arguing about limits?

[1]  This alone tells me I must be getting somewhere in my professional career because typically I just leave seminars more overwhelmed than anything else and have no clue where to even begin to apply the knowledge I just received.

[2]  No piecewise functions yet.

[3]  I understand #2 is a subset of #1 but I find it helpful for students to consider the three different options of zero appearing in the numerator, denominator, or both.

Support

"Little Friend"
by Leonid Afremov

As we're nearing the finish line this year and as I reflect back on my classes, I can truly say that, overall,  this has been a great year.  I've had the luxury of looping many of my PreCalc kids into Calculus (which has resulted in the best class dynamics I've ever gotten to be a part of); I've had amazing opportunities to start to become a teacher-leader at my school; and I've grown much closer to several of my colleagues.  All in all, a wonderful year.

But, of course, it's had its hiccups, too.

I've encountered experiences this year that I knew were eventually bound to happen, but that I've had the fortune of not having to deal with the past five years of teaching.

Experiences that make you question, "Is all this work really worth it?"

Is it worth it when the very people you are trying to help turn against you?

Is it worth it when you can't see the fruit of your labor?

And in those moments, I've found one thing to be true:  the only way to get out of that place of loneliness and despair is to "stand on the shoulders of giants," as our calculus hero Newton said.  I've had to let my guard down, be vulnerable, show my weaknesses, and say to my friends, mentors, and colleagues, "I'm drowning.  Help me remember why it is I do what I do."

And they have, consistently, come to my rescue.

Growing up, my church had this saying, "Faith is a journey, and it was never meant to be done alone."  That's so true, but I think the quote can be made even broader by replacing faith with life.  Life was not meant to be done alone.  Yet, I think teaching is perhaps one of the careers most susceptible to isolation.  It's so easy to focus on MY kids and what we're doing in MY classroom.  It's so easy to stay in my little comfort zone all day and never really interact with adults on a deep level.

And if I choose that path, there are at least two consequences of which I am certain:

1. I will never become the best teacher I can be because I will never learn from those whose strengths are my weaknesses.
2. I will be totally alone in the low moments of this career...which could ultimately lead to burn-out.

I was recently sent this article and I have clung to Glennon Melton's words:

You do not teach by teaching- you teach by loving. Be humble and courageous.

That juxtaposition of humility and courage really hit home.  To be both meek yet bold...that would solve so many of my problems.

And, yet, while it sounds nice, and while it's certainly something I strive for, I realize that it's not something that is just going to happen.  And it's certainly not something that will happen without the help of a community.

So, here's the thing I know:  We have to have a support system.  We have to have people we can call, text, email, Skype, whatever in our dark moments and say, "I freaking suck at this job.  Please help me remember why it is I do what I do."  We have to have people whom we know we can trust.  We have to have people in our lives who look out for us; who put others' needs above their own.  We pour our hearts and souls into the lives of kids, and if we don't have people doing the same for us, we will eventually show up empty, tired, and alone.

We can't be there for our kids emotionally and mentally if we don't have people who are there for us.

After this year, I now know for certain I have people who have my back.  People who want to see me succeed.  People who can restore and replenish me so I can restore and replenish my kids.

I sincerely hope you do, too.

Truce

I was recently at a meeting where two wonderful English teachers were presenting a really cool strategy that they use in their classrooms. They even went a step further and tried to imagine what their strategy would look like in a history, math, or science classroom. Whoa! I thought that was so admirable.

But...they prefaced their math and science ideas with, "We're just really not any good at math and science, so we did what we could."  (I thought their math and science ideas were fabulous.)

*****

At church on Sunday, one of the pastors was talking about how we could help send our teens on spring break missions trips.  He was excited because someone had offered a matching grant for up to $35,000 (wohoo!).  He started giving examples of different ways we could get to that $35,000 mark (i.e., 700 people giving $50; 35 people giving $1000; 35000 people giving $1, etc).  He made a comment like, "My high school teachers would be amazed that I could do this math for you."  Everyone chuckled.

*****

If you're a math teacher, you probably get just as discouraged as I do when adults say things like "Math just really isn't my thing." Or, "I was never really any good at math." I get anxious every time someone asks what I do or where I work because I know exactly where it's going and how it will end:

"I'm a teacher."

"Oh wow! What do you teach?"

[pause here as I wonder if there's a better way to say what I'm about to say...]

"High school math."

"Oh god. I was never any good at math. Glad there are people like you out there."

*****

"People like me"? What does that even mean? What does it mean to be good at math? Whose definition are we using? And please don't tell me you say that in front of your kids because you just gave them a pass not to try in math class.  

As "math people" it's really frustrating for us that it's culturally acceptable to declare your mathematical illiteracy, but totally unacceptable to say, "I was never any good at that whole reading thing."

This double-standard has got to stop. Especially if we're serious about developing our STEM programs around the country.

But...

If I'm being totally honest, I perpetuate that double-standard, too (just the other way around). When kids come and wallow about an essay that's due in history or the novel they have to read in English, I totally play along. And that's totally not ok. Even a simple, consolatory "I'm sorry," should never come out of my mouth. Because I'm not sorry. I'm so glad they're reading and writing and learning about other places and cultures. But I completely admit to not supporting other subjects like I should because I constantly feel the need to defend mathematics.

And so, to the great Internets, can we call a truce? I will be an advocate for your subject and your passion. Would you be an advocate for mine? Can we be a unified front on this issue? Can we vow to defend all aspects of education?

I think our kids need and deserve that from us.  Besides, who ever said you have to choose between math and English, anyway?

The need to teach creativity in mathematics + FRACTALS!

I really resonated with Sam's most recent post about building time and space into curriculum to let students play with math. Mathematics is incredibly creative and innovative and I know that I (and I'm guessing others?) don't take enough time to let kids tinker. It's only when we sit there and tinker (preferably with something that intrigues us) that we become really good at something. This is a truth I constantly try to convince both kids and adults of: a mathematician wasn't miraculously born a "math person"; she found some kind of math that interested her and played with it for a very long time.  I.e., she had to work for it, but she most likely enjoyed the work.

And I am convinced every person can find some kind of math that he enjoys.

And I'm certainly convinced I can do more to be an advocate of the CREATIVITY needed to be successful in mathematics.

So here's a small first attempt! We're currently in our chapter of sequences and series in Precalc (which also includes the Binomial Theorem and hence Pascal's Triangle !!), so I thought it'd be fun to talk about fractals:

Fractals

(Email me if you would like the SMART Notebook file.)

I was able to find a decent video of the Mandelbrot Set (I muted the audio and played it on 2x speed).  We watched this as we talked about what they noticed/wondered.  They were very quick to point out possible fractals found in nature.

And we also watched one of Vi Hart's fabulous clips to motivate drawing fractals by hand (which is pretty darn addicting, no matter who you are):

After this video, we went over how to draw Sierpinski's Triangle (and how it's related to Pascal's Triangle!) and the Koch Snowflake. Then I let them research other fractals (there's a QR code to a Google doc I made with various good links for instructions on how to draw some fractals). Their assignment was to submit a fractal that they've drawn before spring break (either their own fractal or one that's already been "invented"). They are also to include both a recursive and explicit formula that models their iterations.

The types of things I saw as I walked around and the kinds of questions they were asking and the stuff they were pulling up on their phones made me so very happy. I felt completely justified in taking this time to breathe and to play and to create. I think I've said this before, but when you combine teenagers, art, and mathematics, you're bound to be continually impressed. I need to do this more.  

With that said, hopefully I'll have some good pictures to share next week!