## Tuesday, March 12, 2013

### Trashketball (2013 Pi Day task)

It all started with an episode of Suits on USA Network from January 31, 2013 (episode 213: Zane vs. Zane) where the opening scene has the two main characters (Harvey and Mike) playing a round of H-O-R-S-E trashketball in Harvey's office.  I jotted this one down on my digital "task ideas" list and knew it might have some potential later this year in Geometry. Here's Act 1:

Act 1: video to wonder and notice about
Act 2: teacher notes, and visual data/information to help solve the task
Act 3: visual confirmation of the practical answer
Sequel: additional tasks to explore (especially for early finishers) and teacher notes

I was going to chip away at this task until I realized Pi Day was coming up. Needless to say, I started working a little quicker. Ironically, in calculating the answer to the task, Pi can actually be divided by itself or "cancelled." I grabbed (bought, not shoplifted) two trashcans from Bed Bath & Beyond. I found the exact trashcan from Suits. Woohoo!!!! That circular truncated cone trashcan is so dreamy and transparent. I also found a cylindrical trashcan for my Geometry class. As you can see from Act 1, it's not transparent, but it'll get the job done. Measuring each dimension of the can was simple. Measuring the diameter of the trashketballs is a different story. I'm open to suggestions here. You'll find this in the "Teacher Notes"
How do you find the diameter of a trashketball? Have your students come up with ideas. Test those ideas. Make conjectures.
I crumpled up 8.5"x11" paper and made it as compact as possible. I took a handful of trashketballs and put them down on a ruler to get a rough mental mean of the diameters. Then I traced the best-fitting circle to measure the best-fitting diameter of each trashketball. I took the mean of these five diameters.
Seriously, I'm open to ideas. I quickly discovered that trashketballs are like snowflakes: no two are the same. However, I really started to perfect the form and process of making a trashketball. I'll admit, there's some buy-in with the trashketballs being perfect spheres. I'm okay with that. So maybe spend some time with your students perfecting the trashketball. Anyway, leave some ideas about measuring the diameter of the trashketballs in the comments, won't ya?

I'm looking forward to this task. My students occasionally play trashketball in my class with their scratch paper or class handouts (not necessarily mine) contributing to their idea of going paperless. I see this happening a lot on Thursday. Happy Pi Day!

Next up! The circular truncated cone trashcan. I'll start chipping away at having enough trashketballs for the circular truncated cone. Thanks in advance to the following people for helping with the volume of the circular truncated cone trashcan:
@mjfenton, @absvalteaching, @MaryBourassa, and @RobertKaplinsky.

Swish,
1140

1. This is definitely a fun application of volume. In terms of measuring the diameter, and accordingly, finding the trashketball's diameter, it may be worthwhile for everyone in the class to create a trashketball and measure the diameter. If you then average all of their measurements, I would guess that you would have student buy in as to where the measurement came from and it will be reasonably close to what your mean was when you calculated it.

Regarding calculating the number of trashketballs that fill the can versus the volume of the trash can, I just recently had some experiences in a similar type of problem. I was working on the giant gumball machine problem to do with some Geometry teachers. The gumball machine says it is filled with gumballs that are diameter .92" to 1". My initial thought would be that it would be about a one-to-one ratio of gumballs to cubic inches. However, I soon realized that I was mistaken.

First, both your problem and the gumball problem involve "sphere packing" where more spheres can fit in between the gaps. Second, while the gumballs had a diameter of 1", their volume was closer to .5 cubic inches (4/3*pi*.5^3).

Going deeply into the direction of why the answer is obviously beyond the scope of the lesson, but it is definitely the basis of a good discussion.

1. Hi Robert,

Thanks for the comment. I'll give an update on how the lesson went. I added some notes to the 101qs lesson, but more importantly I planned on having students make their own trashketball and that's exactly what we did. Some students also brought up the idea of "sphere packing." Stay tuned for an update. Thanks for the feedback man.

2. Robert: I now see where you got the divide by 2 concept in the gumball. It makes a lot of sense to me now why that is. But, why would dividing by 2 throw off your answer tremendously with trashketballs?

Discussing this image after deriving the formula for a sphere from a cylinder would also be cool.

http://40.media.tumblr.com/9230046fc0b2b32b51a36d715ac22e46/tumblr_n7473u9COK1qzicj3o1_1280.jpg

I'm interested in trying out some 3 acts stuff. I was wondering if you could help me out. I'm curious if how you use these problems. Are you in the beginning, middle, or end of studying volume and you give a problem like this or is it out of the blue, "Today were going to do this cool problem, tomorrow we'll go back to doing ____"?

sam

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