Thursday, March 24, 2016

Volumes Performance Task: A Love Story

So I just need to brag about my kiddos for a second. As I've mentioned before, I teach a Calculus class that is totally new to our school this year. The idea is to give kids 90 minutes per day for an entire year to master AB Calculus. We spend lots more time on practice and on exploring concepts than a traditional class would because we have double the time. Test (and frankly math) anxiety is still a very real thing for some. We just took our volumes test yesterday and I wanted to do something fun on our half day before spring break. I loved the idea of a performance task for these kids so they could jump in and show me what they know in another way, while engaging with each other and using some creativity. 

I gave each group a choice between the clear vases or the colored vases pictured here.

The clear were worth up to 10 points extra credit and the colored were worth up to 14 since they were more intricate. For every 10 mL that the group was off from the actual volume of the vase, I would take away one of those extra credit points. I gave them 35 minutes to plan and calculate, requiring that each group sketch a graph and show their calculus work. 

Then we tested. And it worked

I'll be honest...I expected this to go way differently. I did this activity with a bunch of math teachers (or more accurately watched as the veterans took the lead) when I was just getting back into my calculus groove and planning for my course and we overflowed the crap out of that vase. By my standards, I don't think we would've gotten any points. But my kids were amazing!!! They used all different approaches- some did Riemann sums to find the volume, some used regression to figure out a function that matched closely and then integrate that, others broke it into piece-wise functions and integrated each. They used calipers to measure exact diameters and used a light table to get an exact trace on a graph.  I left it totally open ended and they all took a different approach- but every group nailed it! Each group was within 50 mL of the actual volume- and were so excited to see who would be the closest. I never told them I knew the actual answers, we just tested by pouring the water into the vase.  We talked about over and underestimates. We talked about washers vs. disks and trying to account for the width of the glass. 

Such a rich 40 minutes to send us off for a week and a half. Happy spring break, everyone! 

Tuesday, March 22, 2016

What's a Parabola, Anyway?

One of my favorite conversations with my students is always when we discuss the geometric definitions of conics. It stems from my middle school teaching years, where I would put a bucket of candy in the middle of room and say "first one to get to it wins!" Inevitably, there was always a riot thrown by students who were sitting in the back and felt they were being treated unfairly.

My challenge to them was always this: "Okay...make it fair."

I would watch them as a class try to negotiate how they should stand. Almost always they would try to stand in a straight line in front of it, until the people at the end complained that they were too far away. Someone would suggest making an oval and eventually they'd maneuver themselves into an almost perfect circle, being sure to chastise anyone who was trying to inch closer than the perfect radius they'd formed. So what's a circle? The set of all points that are an equal distance from a center....or a really valuable bucket of candy. 

This idea of identifying the locus of points defined by a certain characteristic carries over in such a beautiful way into conics. I tried something new this year with parabolas (that admittedly I thought of in front of my first period today mid-class) and it seemed to work well and made finding focus and directrix afterwards much easier. 

I started by putting down painter's tape in the hallway to form a line and a point (and build some buy in with the kids because who doesn't love a field trip to the hallway?)

From there I picked a student and asked them to put a cup down at a point that was exactly equidistant from the line and the point. Inevitably, they choose the vertex (which we don't name that....we call it "Jimmy's Point" if Jimmy put the cup down there). Perfect start. 

Next, I challenge another student to do the same, without stacking the cup on top of the other. First try is always to put it right next to it. You better believe this stirs an argument in the group. Someone always runs for the meter stick since we can't just count tiles now. We discuss which it's actually closer to and the class gives suggestions to that student of where to re-position it. I continue giving each student a cup and letting the class hash it out until we've started to notice a pattern. 

It's a parabola! 

We discussed the idea of the focus and directrix at this point and talk about why Jimmy's cup was so important, finally assigning its proper mathematical name. It gave us a great, hands-on experience to tie our knowledge to the rest of class and I saw a big difference in how comfortable my kids felt finding focus and directrix, even from standard form.  

Thinking of turning this into a small group exploration next year where kids can then use the cups to actually do some measuring and explore some traits like focal width or width of the parabola! If anyone has done anything like that before, please feel free to share! 

Saturday, March 19, 2016

Playing Around with Volumes of Revolution

It's finally here...our final topic in AP Calculus!! With the time before spring break winding down, I am wondering where this whole year went and am amazed at how hard my students have pushed themselves. Calculus isn't easy and the course I'm teaching this year is an AP/Honors combo class to help give access to Calculus to students who might not otherwise have felt prepared to take it.  Just wanted to share a few things we're doing this unit in class to practice and cement understanding for the kiddos:

Get the Picture! 
Visualizing 3D volumes is tough. I started the conversation with my kids by looking at some party decorations I'd bought at the dollar store. We looked at the folded shape and the kids predicted what shape decorations I had purchased. This was super intuitive for them and made them start to connect the idea of volumes caused by rotation. 

We also used these to discuss the cross sections that would be formed. Specifically, we talked about a pineapple decoration.  When we slice a pineapple, we know we get circular slices. Duh, they come in a can at the grocery store! That gave us a great frame of reference for discussing why all the cross sections are circular. I'm looking forward to using pineapple slices to talk about washers too- where you can find the whole area and then subtract the area of the core. And anytime I can talk about food, it's a good pedagogical move in my book.

My wheels are turning for a project after the AP exam:

  • Use Desmos to design a 2D shape and determine an axis of revolution
  • Calculate solid of revolution's volume
  • Actually build 3D solid honeycomb decoration (This video is making me think...)

More to come on this later! 

Washer or Disk?
As I mentioned before, my calculus class is an honors/AP full year combo. I have double the time to teach the standard AB Calculus curriculum, which gives me lots of time for practice activities and conceptual explorations to make sure students are understanding. I am always surprised by the concepts with which my kiddos struggle....the topics in which I anticipate difficulty are sometimes easy for them and it's often the background skills that shouldn't be threatening are hard. I am anticipating some amount of practice needed on distinguishing between when to use washer vs. disk method. I think they'll really need to think through and visualize the areas of a few to see the differences clearly. I designed this activity so the kids could sketch the regions for each graph, sort them by the methods used, and then actually calculate the volumes after I'm sure they understand why they would use each method.

Volumes Performance Task 
I've mentioned this task before and I have to tell you that I LOVE it! Our last day before spring break is a half day and what better way to fill it than throwing in some competition and a real-life performance task? 

The basic idea is that students are given a vase and need to calculate the volume of it using calculus. I bought a variety of vases from the dollar store that students will need to analyze. 

The link above is the original performance task that I first tried at NCTM Regionals this year. I will be modifying it slightly and plan to write about my modifications and how it went in my room after! 

I'm excited to get the kids thinkings. They can do this with vertical or horizontal slicing.  they can think in terms of Riemann Sums. They can treat it as a disk or as a washer (if they're analyzing the thickness of the glass). And, at the end of the day, we get to actually test their results and see how accurate they were. It'll a fun way to send them off to spring break and a fun last activity before I hand them their enormous AP review binders. 

Should be a fun last week of content! 

Friday, March 4, 2016

Be More Dog!

I just got back from the North Carolina Association for the Gifted and Talented annual conference and it was such a rewarding experience. I always love meeting like-minded people and pushing myself to learn more....both of which I got to do at the conference. While I have lots to blog about from it, I had one thing really stick out to me as both hilarious and poignant. One of my favorite professional conversations is always that of taking risk in the classroom and while at the conference, someone mentioned this advertisement as an example of that:

Be more dog. It's brilliant.

It's exactly what we need to do in our classrooms every day. It's easy to be "aloof until lunch" and get stuck in our routine if it seems to work for us. However, if we never take a chance on something different, we will never discover all the amazing things that are out there for us- whether that is chasing cars or chasing the perfectly differentiated lesson!

Needless to say, I'm adding this as the very first slide in my presentation for a conference next week and will probably kick off any of my professional presentations with it in the future. Love it!

Wednesday, March 2, 2016

Logarithm Clothesline

I have some typical activities that I use when I introduce logarithms (logarithm "war", Kate Nowak's great intro puzzle, laws of logs explorations, etc), but this year I was looking to challenge my kids and get them talking a little bit more. When I taught middle school, I always loved a good clotheslines activity. It made kids look at multiple representations and analyze them, all while interacting and getting some energy out. I decided to try it out this year. 

Stage 1: Get Them "Designing"

My juniors were out at the ACT, so I had a class of about 8 sophomores (who are obviously very accelerated in pre-calculus) for 90 minutes that I needed to amuse. As a part of that, I gave them the activity below. In it, I gave them the "answers" and they needed to design logarithm questions to match. They got WAY more into it than I thought they would and it actually turned into a pretty engaging activity on it's own

Stage 2: Clothesline

I took the outstanding questions from my sophomores and made them into a sorting activity for the clothesline. 

When kids walked in, I had clotheslines up through the center of the classroom, knowing I'd be putting them in small groups to have them work. I handed each group a baggie with the sort cut out and enough clothespins. I then instructed them to work as a group to order them (which was more of a struggle for some groups than others). 

What I liked most about this was that they actually had to think about a logarithm having a value- something which is usually a struggle for them. They had to work collaboratively and wound up answering a lot of each other's questions before I could. I always love when kids can answer each other in a different way than I can. I had another activity ready for early finishers, but some of them really engaged in it and took the whole time. 

Here was some feedback from the kids. Besides that they wanted candy as a prize and to work with whoever they wanted, they seemed to really enjoy it!