Triangles by side

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

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

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?

Parabolic Telescopes

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

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

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?"