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.

Washer vs. Disks Sort

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. 

What's the Volume Performance Task

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!