From: Ronald Wong (ronwong@inreach.com)
Date: Wed Feb 06 2002 - 11:43:14 PST
Message-Id: <l03102800b8869d3a25cb@[209.209.18.198]> Date: Wed, 6 Feb 2002 11:43:14 -0800 From: Ronald Wong <ronwong@inreach.com> Subject: Re: South Pole Longitude & Compass Question...
"Raleigh McLemore" <raleighmclemore@yahoo.com> said:
>Thanks to everyone for the through answers. I suppose,
>working with elementary school students, that I will
>attempt to interpret this so my students can make
>sens,e of it. I took some string and wrapped it around
>different latitudes, cut it and then stretched it out....
Ahhh, the string....
Now that I know your grade level, I can offer you a fifth solution to your
question regarding the length of 1 degree of longitude when you are near
the pole.
Years ago when I use to do mini-activity/demo in the elementary classroom,
I carried around a collection of cylinders of various diameters and had the
kids use string/masking tape/student-made paper ribbons/... (i.e. whatever
was on hand) to compare the circumference of the cylinders with the
diameter of their bases. As a result, they discovered that no matter how
big around the cylinder was, it's circumference was always "3" times larger
than it's diameter. Amazing!! (n.b. we didn't measure anything in the
traditional sense. The diameter of the base was being used as a unit of
measurement. Measurement itself was another mini-activity/demo I did with
the kids)
Since the diameter was just twice the radius, we could also conclude that
the circumference was "6" times greater than the radius (both of these
concepts having been arrived at through hands-on activity also).
If the students were using calculators already, I'd point out to them that
it's actually a little larger than "6" --- 6.3 to be more exact (correct to
two significant figures - a typical elementary school child's level of
precision). This allowed them to check their discovery using cylinders that
they hadn't considered before using actual numbers on a measuring tape
(mugs, glasses, chair-legs, anything we could find around the classroom).
Generally, we'd measure to the nearest inch. This way they became
comfortable with dealing with things by way of approximations and less
focused on "exact" answers.
We then went on to see how we could use our new found knowledge to get the
radius/diameter of a sphere - something that would be more difficult to
measure without destroying the sphere (out came the spheres!). After that
they'd start measuring things like their heads, arms, and so on. Well, you
know what it's like with a classroom full of elementary school children.
At some point I would ask them if they could think of a way to measure the
diameter of their spheres without measuring the circumference. Once they
figured out a way, they were able to check their predictions of the
diameter of the spheres based on the circumference they had measured
earlier. Great fun.
What has this got to do with the size of 1 degree of longitude near the pole?
Well, when you had that globe, you could have rotated it so that the axis
of rotation of the earth was pointing at your students. They would
immediately see that the parallels of latitude form concentric circles
centered at the pole (approximating a polar orthographic projection for
those of you who are searching for some big words to throw around).
If they were aware of the "Rule of 6.3" (easy to remember: there are 2
digits and, when you divide the larger by 2 you get the smaller - they'll
never forget that the larger is 6. Hopefully, they'll be able to figure out
that it must be the radius that we are dealing with here and not the
diameter), then they would just take an arbitrary distance from the pole,
multiply it by 6.3 and divide it by 360 degrees (remember: the circle of
latitude is 360 degrees of longitude and we only wanted the length of 1
degree of longitude). Voila! The answer to your question.
To use the previous example at 1 degree of latitude from the pole (i.e. at
a latitude of 89 degrees north or south), you just have to know that a
person one degree of latitude away from the pole is 69 statute miles away
from the pole.
(6.3 X 69) / 360 = 1.207 miles = 1.2 miles (the factors have as few as 2
sig.fig. so the answer has only two). That's the fifth time we've come up
with this number.
So, at last, we have the simplest solution to the problem Raleigh posed (IF
the students are familiar with the "Rule of 6.3"). To paraphrase Mr.
Holmes, "Elementary, my dear Raleigh!"
ron
p.s. Why a person 1 degree from the pole is 69 statute miles away from the
pole involves another area I've considered for an elementary school
mini-demo: "The silliness of the British Poundal System". It never ceases
to amaze me how many people come up with the wrong answer to the question,
"Which weighs more, a pound of feathers or a pound of gold?" and it's
companion, "Which weighs more, an ounce of feathers or an ounce of gold?"
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