Coffee into Theorems

Triangles by side

2009-10-31T11:43:00.000-07:00

I just used a mathlet I found online: "Triangles By Side" (source: Math Hombre) in a lesson for geometry. (uploaded here: http://scottfarrar.googlepages.com/geom2009 The worksheet http://scottfarrar.googlepages.com/TriangleCategories.pdf ) It worked very well.

A few notes:
0. This was our second day on triangles so I had not used the vocab for each category. I purposely left it to the end of class then we all labeled each category as a class. Students worked in pairs, 1 computer per pair.
1. Students were a little confused about starting with the scalene. There's not much to "observe" there. I think the best one to start with is the isosceles.
2. Its kind of a shame that the 3,4,5 and 6,8,10 are the only right triangles possible. Next time I would bump the sliders up to 13 so that students could make a 5,12,13. (obviously we haven't covered pythagorean, but students were able to find the triangles easily enough)
3. I actually didn't have the 2nd mathlet up there when my students did it. They used the first one for both activities.
Pros: integer lengths for c are easy to list. Students had no trouble figuring out what to do.
Cons: students don't automatically consider fractional side lengths for side c.
This can be a pro, however!! A kid says if a=4 and b=6, c can be 3, 4, 5, 6, 7, 8, 9. Then they are ripe for me to ask "can c be 2 and a half?" They can flip back and forth from c=2 and c=3 to guess at what c=2.5 looks like. Then I ask "can c=2.1? 2.01? 2.001?" It was great to have students interrupt me half-annoyed and say "As long as its more than 2, its ok"

The one I just made (the 2nd one for 10-30) might be "too helpful" for day 2 of triangles. This is probably better as a review or lecture demonstration. http://scottfarrar.googlepages.com/triangleineq.html

So I think I might change my worksheet back to using the first mathlet, or a modifed version of the first. I'd limit the way they interact with side c first. Then I can give them more freedom to explore rational side lengths.

I welcome feedback and suggestions!

Circumcenters and Epicenters

2009-10-13T21:48:00.001-07:00

http://scottfarrar.googlepages.com/circumcenter_lomaprieta.html

So I put together this mathlet in anticipation of doing Triangle Centers. I love it, and I hate it. On one side, I'm very satisfied with how it turned out implementation-wise. On the other side, I'm not satisfied with what the lesson is. This is not a 50 minute activity. So they find Loma Prieta. Big whoop! There's not really a *problem* to solve.

Could I muddy up the data? Could I muddy the data? Or should I go and try to get actual USGS data in terms of when the first shockwaves were felt and where. With all the differing topography in the Bay Area, I'm sure the shockwaves were not perfectly circular. And yet, if we took a lot of data we could probably do the calculations/constructions needed in order to find a good estimate for Loma Prieta.

Whats frustrating is that this lesson is not ready, and I'm not sure if I'll have time to get it all the way there by the time this comes up in the year.

Introductory Geogebra Lesson

2009-10-10T14:04:00.001-07:00

I did this with my students about a week ago: http://scottfarrar.googlepages.com/geom2009

Previous Knowledge: students have copied and bisected angles and segments before "IRL" using compasses and straightedges.

It went very well for a "first time" on computers this year. The versions of the files I initially used did not limit their tools, but I have now changed the mathlets: you are limited to Euclidean constructions via compass, straightedge and points/intersections.

The fourth mathlet is too hard. The first three took most of my students about 20-40 minutes to get through. So I definitely need a fourth problem that is relatively simple, yet exposes them to something new in Geogebra.

"Free exploration time" works for some students, but if they are that interested, they can do it at home on their own computers. I'd rather have an engaging mathlet.

Is there a Geometry lesson here?

2009-09-18T17:53:00.000-07:00

Parabolic Telescopes

2009-05-31T16:04:00.000-07:00

Last week as I was going through a whirlwind tour of conics to wrap up my Math Analysis class, I wanted to illustrate some real-life examples. Rather than just saying "these things are parabolas, these things are ellipses [etc]" I wanted to have the kids DO something with conics.

I'm not sure I succeeded, but I did come up with one activity that, with further work, could be a nice problem for future Advanced Algebra / Analysis classes.

I had a nice high quality picture of the Very Large Array in New Mexico, since my sister and I visited there on a cross-country drive. I wondered if it was possible to find an equation that would actually model the parabolic shape of the telescopes. So I popped the picture into Geogebra and constructed a parabola via the locus tool (which I just recently learned how to use).

The stated goal of the assignment is "find an equation for the telescope." I told them not to worry about the rotation. We could handle that later. (we didn't handle it, but perhaps next time, if this comes after matrices we could multiply by a rotation matrix... find the angle by inverse trig based on the slope of the dish)

But I think if I had used Geogebra with the kids more this year, they might have the capability to construct the locus themselves instead of me giving them focus / directrix. Basically, if they did what I did, I feel they'd get a good understanding of what a parabola is from a locus standpoint.

I didn't put this on the mathlet, but the diameter of one dish is 85 feet. Two ways to handle that: (1) rescale it so the geogebra numbers match it. (2) have the kids find the scaling factor (put points on the locus, measure the diameter using a segment). I like option 2, but that does require them to be well versed in Geogebra (goal: verse my kids in Geogebra next year!)

Well, I appreciate any comments / critique.

http://scottfarrar.googlepages.com/VLA.html

f(t): Help Me With Some Algebra

2009-03-21T17:53:00.000-07:00

f(t): Help Me With Some Algebra

Given a chord AB and the intercepted arc S, is it possible to find radius r algebraically? (problem stated at f(t)) The consensus there seems to be no, due to the transcendental nature of sin(x).

But I find it interesting that r could be constructed with a compass and straightedge. However, I feel like I'm cheating with Geogebra since I had to place C in order to draw arc S. So I really wasn't given S; I picked one based on C. Circular logic! (that's a little math joke ... ha ha ha.)

Update: http://www.mathforum.com/dr.math/faq/faq.circle.segment.html#1 Dr. Math has a nice page about solving circles given arbitrary parts. They reinforce the idea that this is a problem with numerical-only solution. Their solution involves Newton's Method, something I myself am not very well versed in these days!

News Media

2009-03-19T23:23:00.000-07:00

This XKCD comic reminded me of why I hate most of the News Media. The panel on the left is a pretty accurate rendition of a full screen graphic one of the 24 hour networks had up:

AIG
Bailout: $170 billion
Bonuses: $165 million

At first glance, one might arrive at the conclusion that AIG was using almost its ENTIRE bailout for these bonuses. "How irresponsible! What an outrage! Despicable!" But upon closer inspection, one notices the difference between the numbers. What if we actually took the difference?

$170,000,000,000

- $165,000,000

= $169,835,000,000

There's $165 million is a drop in the hat. Its about 0.1% of the bailout money. It pales in comparison! It should not be compared. I'm not defending AIG. That's not the issue. The issue is the News Media is presenting information in a misleading format. I also assert the News Media is doing so purposely. They want the outrage, because outraged viewers will watch more. Everything is sensationalized and distorted.

The new propoganda isn't for political gain. The Media wants your viewership and your money. They will distort the facts for it.

Our financial crisis is a result of people misrepresenting information about loans rates, feasibility of real estate transactions, investment options and the market in general. People would bend the truth to gain more money. Now, the News Media is now misrepresenting information... to gain more money. They mimic the actions of the very organizations they codemn.

"Why do we have to take math?" students ask me. Look at the world. Those who understand the system will take advantage of those who do not. A simple concept like orders of magnitude (billions vs. millions) will be lost on many viewers who have been in the habit of non-mathematical thinking.

The next time stats or data is thrown in your face, by the Media or anybody else, pause and think: "what does it really mean?"

f(t): How to Bounce a Ball Part 1 - The Problem

2009-03-13T02:31:00.000-07:00

f(t): How to Bounce a Ball Part 1 - The Problem

A student-led lesson (Geometry)

Barackroll and Rickroll

2008-08-24T11:24:00.001-07:00

full circle

Wikipedia Crawl

2008-01-03T02:40:00.000-08:00

I was channel surfing and stumbled upon The Late Late Show with Craig Ferguson. He was doing a Michael Caine impression and made some joke that I didn't quite get. Wikipedia to the rescue! or so I thought...

1:36 am
Michael_Caine
Scottish_English
Regional_accents_of_English_speakers
American_English

1:58 am: I split into two simultaneous tabs:

Branch 1:
Regional_vocabularies_of_American_English
--#The_West
Jefferson_State
State_of_Franklin
State_of_Lincoln
List_of_U.S._states_by_date_of_statehood
Historic_regions_of_the_United_States
--#Unrecognized_or_self-declared_entities
2:24 am

Branch 2:
American_English_regional_differences
California_English
--#Freeway_nomenclature
California_vowel_shift.png
Hyphy
2:24 am

There are some real gems in there. Never did find out what Ferguson was referencing though.

xkcd #214.

iPhone

2008-01-02T20:19:00.000-08:00

What a piece of work is iPhone, how noble in reason, how infinite in faculties, in form and moving how express and admirable, in action how like an angel, in apprehension how like a god.

Hamlet was one of the first people to get an iPhone but I was only 6 months behind! Scientifically speaking, it rocks.

Using elementary set theory, I will describe precisely every way in which it rocks, by articulating the complement.

{How the iPhone rocks}^c

No Copy/Paste. Well, no text highlighting in general. These are essential tools, and I feel handicapped without them.
No GPS. This doesn't sound so bad to most people, ("why do I need to know exactly where I am?") but once you start playing around with the contacts and google maps then you realize you'd want some sort of functionality of "how do I get from HERE to there?" Basically, putting in GPS would have made the iPhone the closest thing to a URAT* modern society has to offer.
Needs a search filter on Contacts. As fun as it is to flick my contact list to scroll through them all, I would like some sort of shortcut to get to a specific person.
EDGE is slightly slow. But frankly, it works just fine for most of my needs. Even downloading stuff from YouTube isn't bad with it. I sometimes even turn off Wi-Fi to go to Edge when the network signal is weak.
Calendar only syncs with Outlook 2003 or 2007 on Windows. I would much prefer it to sync with Google Calendar directly. You can access Google Calendar through the web, of course, but I want to use the iPhone calendar "without all that tedious mucking about in hyperspace."*
Headphone jack is recessed. This is so the curved edge of the iPhone is still curved, and it actually protects the structure of the headphone jack, the internal card, and the plug. But it also means all standard headphones wont work without an adapter... $25.
No 3rd party apps -- yet. Sure I can go through Safari to do web-based apps, but that's an inherently bulky way of doing things, and what did I just say about mucking about...
No direct interaction with files -- yet. I can't see the files I have on there ("the files are in the computer"*) or do anything with them, such as saving a picture off the internet, deleting songs, editing tags, editing documents.

That's it. And essentially, most everything will be great when Apple puts out their Software Development Kit (SDK) and lets Google work their magic. Google people are SMRT*.

Symmetry!

2008-01-02T17:53:00.000-08:00

What do these words have in common?
dollop, suns, noon
Ok, try turning the screen upside down and reading them... (for 'noon', try reading it from behind your screen... oh wait... doh!)

I was over at Tom's the other day. We were talking about Blu-ray and HDDVD and how movies will inevitably be re-released forcing consumers to either cope with the fact that they do not have the pinnacle releases of their favorite movies, or to take cash against a sea of discs, and by conceding, buy them. Sidenote: how much clearer can a movie like the Simpsons get?

Whilst I was making fun of Tom for having multiple copies of some movies in his collection already (Princess Bride, Star Wars... ) he mentioned that there was yet another release of Princess Bride. I was ready to add another $40 onto the running tally (yeah, I keep a tally of close friends' leisure expenditures in my head, you wanna fight about it?) he mentioned how the cover of this latest release was quite interesting.

If you rotate the cover 180 degrees, the title reads the same way. Thus they have created an rotational Ambigram. Just like the words suns and dollop, it is symmetric under a rotation on the z axis. (z positive is coming out at you, z negative is looking directly into the screen).

The Princess Bride, however, is not a natural ambigram like dollop and suns. It took some nice artistic work for that one. Perhaps there is a job for math majors. "Oh yes I graduated with a B.S. in Math and an emphasis in ambigram synthesis."

In reading the Wikipedia article it saw examples of other ambigrams. noon, for example, is also a 180 degree rotational ambigram, but you rotate it around the y axis (between the two o's)

Then there's the rather impressive x-axis symmetry of
HICKOK DIED DEC 3 1883 -- DOC BEECH DECIDED HE CHOKED. Imagine the 1 has no little doohickeys on it. I actually saw this at Asilomar during Harold Jacob's presentation.

Ambigrams have actually become more popular recently with Dan Brown's cover for Angels and Demons. Dan Browns books are nice for inciting a little pop-math fever, they aren't exactly rigorous... but they did lead to a re-release of John Langdon's (he being the basis for the Da Vinci Code's protagonist) book Wordplay.

Anyway, this got me thinking... is this a key to more hidden elements in Princess Bride? The 180 degree flips of Wesley and Inigo's dominant hands during their duel suddenly seem more mathematically significant. This scene is actually called "swordplay/wordplay" on DVD chapter selections. Dun dun dunnnnnnnnnn....

The trouble with Myspace

2008-01-02T15:13:00.000-08:00

This is the reason why Facebook is better than myspace. Too bad Facebook is converging towards obnoxiousnessity with each subsequent app.

The old (circa 2001) computer scientist in me also prefers the object oriented programming feel of Facebook.

p.s. Object Oriented ProgramS = OOPS

Pop culture data representations

2007-12-15T00:18:00.000-08:00

Many people have seen, or even made, drawings or illustrations depicting song lyrics either sincerely or sarcastically. But how many have made accurate data representations of pop culture? I will share with you some examples.

I like the cleverly inactive "Touch this"

Layers upon layers.

Biggie never said anything about it being linear though... he merely asserted that for the function f:money->problems, f'(x) > 0 for all x.

Triangles from folding

2007-12-09T20:29:00.000-08:00

Using a square piece of paper, label each side 1 unit long. Find the midpoint of the top side by folding in half. Take the bottom right corner and fold it up to touch the midpoint of the top side. This creates three triangles. ABC, CDE, EFG. Determine the side lengths of all three triangles.

This was a nifty little problem given by Harold Jacobs at Asilomar 2007.

Prison Break

2007-11-04T11:22:00.000-08:00

Having never seen Prison Break, I have gotten more and more confused about the premise. All of my knowledge comes from FOX's endless promotions of the show. Here's what I've pieced together:

Guy gets accused of a crime he didn't commit.
Brother gets into prison in order to help Guy break out.
Shenanigans ensue.
They break out and meet their father? or something

Now in the new season the Brother guy is back in prison? But, in like China or something?

I guess my main question is, how many times does he have to break out of prison?

Dr. House

2007-11-02T16:51:00.000-07:00

My Halloween costume was "House" from the tv show House. "TV's most liked jerk." My limp was apparently very convincing as one woman offered to help me carry my backpack...

Ok after the pictures sat for a while and ripened on my camera, here's me as house, complete with poorly photoshopped background.

15 Laterals

2007-10-29T22:59:00.001-07:00

So the NFL went to London last weekend, and Trinity of Texas decided to add a little rugby flair to their own game... You can also tell this is division III football by the distinctively "brah" accent of the color announcer.

Laughing Baby

2007-10-28T21:34:00.001-07:00

While we all enjoy this laughing baby now, will Ethan still be laughing when the parents bring it out to show it to his high school prom date?

More baseball

2007-10-28T20:12:00.000-07:00

A-rod apparently just opted out of his Yankee contract. And the Yankees said they wouldn't resign him if he did that. Boston next year! Unless he all of a sudden becomes altruistic and joins the A's...

Swisher 1B
Ellis 2B
Rodriguez SS
Chavez 3B

Now that's an infield.

Coors 1, Sam Adams 4 right now... c'mon Silver Bullets!

Boston the new Evil Empire?

2007-10-28T18:24:00.001-07:00

Are the Boston Red Sox going to be the new New York Yankees? ("New New York" ala Futurama?) Their payrolls aren't all that different these days. Boston's $150 million to New York's $190 million. Despite the fact you could pay the entire roster of both the Devil Rays AND the Marlins with the difference (and have money left over), both teams are of the same mindset: win, literally, at any cost.

The other thing about Boston is all of a sudden they're winning world series(es -- TBD later tonight!) and the Yankees are apparently cursed... being feasted on by a plague of locusts and whatnot. (not quite this bad) It'll be interesting to see how far the Yankees roster shakeup goes. But in reality... what other teams can afford Arod? Boston... great!

Now instead of whiny Bostonians and obnoxious Yorkers we have to put up with the opposite!

An attempt at blogging again

2007-10-28T18:06:00.000-07:00

We'll see how far this gets before I leave this blog to die with the rest. This one is cooler than the rest though, cause I have RSS feeds of XKCD.com and Dinosaurcomics.com.

2007-09-30T21:21:00.000-07:00

e^(i*pi)+1=0