Coffee to Theorems

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!

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.

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.

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