Ok, we've been discussing the square dissection problem for a few days now. The original problem was here and Solution Method I was here.

Now don't start laughing at these crude, low-tech videos! The 1st video is 2-3 minutes long. It provides a quick overview of the squares problem and a demonstration of why the product of the areas is the same for either pair of non-adjacent rectangles. It ends abruptly with part of the derivation I gave in the previous post.

The remaining presentation is split into 3 segments in order to control the file size (Blogger has a problem with larger files). In these segments, I derive a formula for the maximum product using the Arithmetic-Geometric Mean Inequality (AM-GM) and apply it to the square dissection problem. The first of these clips reviews this important and useful inequality, presenting a standard algebraic derivation. The 2nd of these 3 clips presents a geometric derivation and the final clip shows how we can derive a formula, m^{4}/16, for the maximum possible product of the areas in the general case of a square of side length m.

As I say on the video, there's no SmartBoard technology, no Mimio, no whiteboard with vivid color dry erase markers! Just an old chalkboard my kids use in the basement, some inexpensive chalk and sheets of paper towel to erase. I wore my favorite 'Pop-Pop' sweatshirt - sorry, no formal wear.

Let me know if you find this helpful (once you stop chuckling!). At least you'll be able to match a name to a face and it isn't Archimedes! If I get few if any comments, I'll know that my 15 minutes of fame have expired.

## Sunday, December 2, 2007

### The Dissection of the Square Problem - Part III - A Video Surprise

Posted by Dave Marain at 6:01 PM

Labels: arithmetic-geometric mean inequality, square dissection, video lesson

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## 7 comments:

Hi Dave,

I couldn't but help noticing a resemblance to Dennis Hopper.

Also, as we are already in the general subject area, if you called the set of videos "The Aritmetic-Geometric Mean Inequality: An Inconvenient Truth," you may even be able to snag an Oscar nomination :-)

TC

Gee, thanks, tc! Arte you referring to Dennis' mugshot or when he was acting?

Aside from whom I remind you of, what did you think of the

contentof the videos?I am trying to decide if occasional use of video technology would be helpful to teachers or students. There is other technology I'm also considering, which would allow electronic capture of handwritten notes and diagrams from a whiteboard, assuming the file can be put in a format that can be published on the web. More to come...

Dave,

I'm interested in hearing about this other technology you refer to. Any idea when that will be forthcoming?

Nice, Dave, and nice to put a face to a name. :)

I think some solutions come across better in the conversational blackboard mode rather than the textbook mode. Thus, there certainly is a place for the video solutions that you post.

However, these will be more appreciated by students at that level, rather than the teachers who seem to be the main audience for this blog (at least, that is my impression from the commenters). The value to them from a video solution is lower, IMHO, since it takes more time to get the same information that could have been obtained by reading something.

Of course, I realize how painful it is to typeset the equations, and the video method certainly enables you to transcend that aspect, so all power to you!!

I wonder if the other technology you refer to is a digital camera. With current resolutions, even pictures of printed documents are quite readable. There is also a whiteboard sold with an arm that moves across scanning the whole board, delivering a printout at the end. Apart from the novelty, we never found it very useful in my workplace, and the cost was quite high. There is also a pen computer that basically scans as you are writing (I see ads for this this Christmas season).

Cheers,

TC

The alternate technology I referred to has actually been out there for about 10 years. I've used it in the classroom with success since it enabled me to download to my laptop every single mark I placed on a plain old whiteboard. The device digitally captures every mark you make and even every erasure using special dry erase markers and a special eraser. One can then email these notes to students who are absent (I used to do that before I left for the day). Further, this made my life easier when providing student notes for students whose IEP's required them. The technology has improved since then and I will share more detail and perhaps even demonstrate this. What's really cool is that the board can also be captured in animated mode and re-played in a QuickTime movie so that it can be replayed stroke by stroke. This is somewhat different from a SmartBoard.

tc -- I used a digital camera to record these videos, then used iMovie on a MAC to split the larger video file into smaller files, then compressed them into .mov format for Quicktime, before uploading them to Blogger. I could have used You Tube of course, but I wanted to give Blogger a try, since it now allows video uploads. The quality is fair and the streaming will back up if too many viewers are trying to get on, but it's a start!!

I do agree that the presentation would be more appropriate for students who are learning the inequality, but I also believe this is a topic most secondary teachers don't usually get to, nor is it in most curricula. I was also demonstrating some pedagogy here, although without a tc, mathmom, jackie,eric or jonathan in my 'class' to challenge me, it wasn't quite a school-like atmosphere. I'm sure I would have gotten some great insights from all of you if you were physically present in my basement!

I'm experimenting with different ideas for my blog. I may try this again, but I don't need the 'over-exposure!'

I finally got around to viewing. I do like this presentation better. There are little verbal tics and flourishes that can be overlooked in text, but "live" you get my mind racing in several different directions. Also, you speak a bit slower than I read, so I kept listening and simultaneously trying to guess what was coming next. That is fun, but it also stretches me.

Thanks.

jonathan

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