Tuesday, August 15, 2017

Elegant Solutions vs. Hacking

This is a study in contrasts around a fun problem by @eylem:

An elegant solution to this would be as follows:

  • After angle chasing to find angle DEA is 45 degrees extending DE to make a full right isosceles triangle with side lengths of 15 sqrt(2)/2. 
  • Use the Pythagorean Theorem to find the missing side length of AGE. This gives you the base of the triangle DE also,
  • Then note AGE is congruent to the halves of CDEs so you also know the altitude.
  • Apply the triangle area formula.

But there are other shadier options for solving the problem:

Instead, we can take advantage of patterns in the values of the lengths and the then use knowledge about 3-4-5 triangles (see: http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html) to crack the problem.

This actually exposes a bit of structure that was not seen in the first solution.  If you didn't think to try the educated initial guess, you could more directly find it by setting up a simple quadratic equation based on the Pythagorean Theorem:

$$ (5\sqrt{5})^2 + (15 - x)^2 = x^2 $$

Tuesday, August 8, 2017

How I use Twitter

This post started with some musing about the meta conversations occurring online right now in twitter over hashtags.  There was enough activity that I was distracted for a bit and then I came to the realization: This is not why I'm on social media.

I started logging on to Twitter in January of 2015 after seeing some math posts on google+ that seemed to indicate there was a lot of interesting activity going on.  After a few weeks I was convinced:  twitter was a great place to find other people actively discussing mathematics and working with kids.

I started by monitoring #mathchat which has a mixture of spam/ads and real people.  From there I found #mtbos which is much more focused on teaching, has less spam and a more communal feel. Over time, I've been slowly growing my follow list of people who post interesting ideas.  Building this network has tended to focus the tweets and made the experience more useful.  Although compared to average I have both less followers and follow less people which probably reflects my own style for engaging with social media.

What I'm finding over time is there are several different types of posters that I enjoy the most.

1. Puzzle and problem producers. These are folks like @gogeometry, @eylem, @cuttheknot, @five_triangles, @sansu_seijin, @solvemymaths.   They regularly post interesting problems that I like to try out and reuse.

2. Animated Gif Makers. There are a ton of really interesting animated math gifs being produced by folks like @gohio, @dynamic_math, @beesandbombs.

3. Individual Bloggers. These are folks who's blog or channel I regularly read anyway like @mikeandallie, @hpicciotti, @fawnpnguyen, @mrhonnor, @math8_teacher, @standupmaths
I've found a lot of sources for ideas for ideas about activities or pedagogy this way.

4. Conversationalists: These folks are usually less for blogs and more for the conversations they produce in twitter itself. @mpershan, @trianglemancsd

The flip side is that twitter is not all goodness. Its very hard to express complex or nuanced ideas in the limits of a tweet. And then there are viewpoints out there with which I don't agree. I'm definitely susceptible to the "But someone's wrong on the Internet!" phenomena, especially if a person posts a lot.  So I try very hard to filter these out rather than responding most of the time. Arguments in 140 character tweets are not terribly exciting. In fact, there are even a few folks I stopped following even though I agreed with them because most of their time was spent in debates that were not useful for me. In the end, my mantra is try to produce the content that I would also like to consume.