Wednesday, November 16, 2016

Justifications and Curve Sketching in AP Calc

Have I mentioned that I love curve sketching? (Yes, I know, only every 5 minutes). To me, it's a giant puzzle that students can approach from all different angles. There are so many small parts that individually contribute to the big picture and I love those moments when you see it all come together for a student. It's one of the only units in Calculus where I can look around on any given day actually see the lightbulbs lighting up for each kid- each one benefiting from a different perspective or activity. Today we did some serious curve sketching challenges!

#1: Justification Graphic Organizer

Students worked in small groups to complete the graphic organizer with calculus explanations for each of the features in the chart. 

We returned to their chart over and over throughout the lesson and it's something they can keep to study from in the future. Next year I think I'd like to revise it to to include explanations of what happens on f, f', and f''...maybe a sketch or description box?

#2: Quiz, Quiz, Trade

This is an activity that is right from the College Board modules and I love it! I gave the students time to work on their own problem silently and then check their answer. They needed to be prepared to teach it to another student. If they didn't feel prepared, they got to come hang out with me at a side table to go over it once the activity started. I set a timer for 5 minutes and told students to interact with as many other students as they could in that time. They worked diligently, explaining their problem to those who needed help and solving each others problems. It brought out common misconceptions, too (which is always vital).  

#3: AP Justification Challenges

Next we worked on scaffolded activities to get students building arguments. Here we discussed what implications f, f', and f'' all have on each other more specifically and began to see common AP style questions. We didn't finish all of these in class, but the nice thing is you can pick and choose what your kids need to most help with while you work. 

Here's the whole document that we used!  AP Style Justifications

Let me know if you have any other activities you love for this unit! 

Friday, November 11, 2016

AP Calculus Curve Sketching Tips & Tricks

One of my favorite times of year in AP Calculus is that moment when we finally have enough mastery of derivative technique to start seeing the "why" to our last few weeks of "what."  These are some of my favorite conversations I get to have with my kids (and my kids get to have with each other) as we move into curve sketching. 

Extreme Value Theorem (aka Wolf Blitzer's Celebrity Jeopardy Meltdown)
Extreme Value Theorem always seems to fall around election day in my curriculum and the the obvious connection between that and the candidate test are often staring me directly in the face. However, I've found that this analogy often only helps students remember the name of a test- not that actual idea of what it does. 

My husband and I are both secretly 87 years old and we watch Wheel of Fortune and Jeopardy every night together. When I was recently reminded of Wolf Blitzer's total meltdown on Celebrity Jeopardy, I had a sudden ah-ha moment:
  • There are only 3 candidates here- the people on the show. Are there people yelling the answers from their couch at home? Sure. But they aren't true candidates for winning since they aren't even in the studio. Same thing goes with the candidate have to be in the interval to be eligible.
  • While all candidates are eligible, it's pretty clear to see who the winner and loser was in this episode. There's an absolute maximum, an absolute minimum, and another candidate who just gets a "thank you for playing."
  • It's easy to talk about ties with this analogy as well. If someone had tied Andy (most likely not Wolf), there would have been 2 winners (2 absolute maximums). The key is they would have had the same maximum value (y value) occurring for 2 different people (x value). It's an easy conversation to revisit when you are distinguishing between an extreme value and where it occurs. 
Mean Value Theorem(aka the E-Z Pass Conspiracy Theory )
Kids love a good conspiracy theory and what better tool to scare a new driver than the prospect of an undeserved speeding ticket. I've posted before about how I introduce Mean Value Theorem, but this year I added a new layer of intrigue. This Snopes Article linked above was written about the area in which I am teaching (Upstate NY) and I remember being a teenager and hearing family members talking about "big brother watching" with all the new technology that was being developed. I may or may not have given my students an article about this to read before showing them that Snopes had verified it wasn't true. They may or may not have freaked out a little. It definitely got them talking about what type of proof they would want a police officer to present to them if they knew that when they were clocked on a radar gun they weren't speeding, but the cop was giving them a ticket anyway. 

Rolle's Theorem (just think "Tootsie Roll")
My husband still talks about the day he learned Rolle's Theorem in BC Calc. He can't tell you anything about what it meant, but he remembers that his teacher gave them all dinner rolls. I thought about doing this and was talking at a dinner party one night about where to buy dinner rolls in bulk (because math teachers at a dinner party are notoriously cool). After resigning that I'd need to look like a crazy lady at a wholesale club the next day, someone I barely knew suggested I just buy Tootsie Rolls instead. I had another mini-epiphany:
  • Roll sounds a lot like ROLLES Theorem (obvious, I know)
  • Tootsie Rolls look like a horizontal tangent line if you hold them up to read them correctly. Their slope? Zero! 
  • The letter o is everywhere here: rOlles theorem, tOOtsie rOll. And what do o's remind us of? ZERO!
  • You can even use the Tootsie Roll to demonstrate the theorem by sliding it between the tangent and the secant lines:
First Derivative Test- Let's Get Talking
The first derivative test is a pretty intuitive idea when you break it down....a maximum has to happen when slope changes from positive to negative and a minimum has to happen when slope changes from negative to positive. It's something you could talk about at a much lower level than AP Calc. However, it's something we need to be able to justify correctly at the AP level, so the more I can get my students to understand the intuition, the easier this will be for them. I used this activity this year:

I gave out cards like this to my students. 

Each one has:
  • The graph of a derivative on it
  • A prompt to help students generate discussion
  • An answer key on the back.
First, I had students work individually on their own problem to make sure they were the "experts" in it. Then, students "speed-dated" students with different problems, trying to determine intervals of increasing and decreasing on each. Then, in small groups, students were asked to identify what happens at the points where the intervals meet. We wrapped up with a whole class discussion to debrief and develop our final conjecture. 

We still have a lot more to do in curve sketching, but these are just some of my favorite little tricks for introducing a collection of really fun ideas!