TACTics Journal
/bigger>/bigger>/bigger>/bigger>/bigger>A Publication for and by TOC
for Education Practitioners
May 28, 2004
/fontfamily>
/fontfamily>In this weekÕs issue:
/smaller>/fontfamily>Elementary/Secondary
TACTics
/fontfamily>(1) The Earth in the Solar
System, /fontfamily>Mike Round
/fontfamily>EditorsÕ Notes
/fontfamily>(2) /fontfamily>Kay
Buckner-Seal, Cheryl A. Edwards
/fontfamily>
/bigger>/fontfamily>/flushboth>ELEMENTARY/SECONDARY
TACTICS
/bigger>/fontfamily>(1) The Earth in the
Solar System: A Logical Look at Orbits, Seasons, Rotations and Revolutions
/fontfamily>Mike Round, TOCFE
Director, USA
/fontfamily>LetÕs assume the distance from
the EarthÕs core to the outer surface is 4,000 miles. What can we conclude from
this?
/smaller>/smaller>/fontfamily>IF:/fontfamily>
The distance from the EarthÕs core to the outer surface is 4,000 miles.
/fontfamily>AND:/fontfamily>
The distance from the EarthÕs core to the outer surface is the /fontfamily>RADIUS
/fontfamily>of the Earth.
/fontfamily>THEN: /fontfamily>The
radius of the earth is 4,000 miles.
/fontfamily>IF:/fontfamily>
The radius of the earth is 4,000 miles.
/fontfamily>AND:/fontfamily>
The circumference of a sphere is equal to 2/fontfamily>¹/fontfamily>
(radius)
/fontfamily>THEN:/fontfamily>
The circumference of the earth is equal to (2)(3.14)(4000 miles) = 25,120
miles.
OKÉ if the earth has a radius of 4000 miles, then it has a circumference of
25,120 miles. What do we know about the /fontfamily>SPEED/fontfamily>
of rotation?
/smaller>/smaller>/fontfamily>IF:/fontfamily>
The Earth is a sphere.
/fontfamily>AND:/fontfamily>
The distance from the EarthÕs core to the outer surface is 4,000 miles.
/fontfamily>THEN:/fontfamily>
The circumference of the earth is equal to (2)(3.14)(4000
miles) = 25,120 miles.
/fontfamily>IF:/fontfamily>
The circumference of the earth is equal to (2)(3.14)(4000
miles) = 25,120 miles.
/fontfamily>AND: /fontfamily>The
Earth makes one complete rotation every day.
/fontfamily>THEN: /fontfamily>The
Earth rotates at a speed of 25,120 miles / 24 hours.
/fontfamily>IF:/fontfamily>
The Earth rotates at a speed of 25,120 miles / 24 hours.
/fontfamily>THEN:/fontfamily>
The Earth rotates at a speed of 1,047 miles/hour.
Examination of that conclusionÉwhat are the implications?
The Earth rotates at a speed of 1,047 miles/hour.
/fontfamily>THEN: /fontfamily>Everywhere
you are on Earth, you are rotating at a speed of 1,047 miles/hour.
HOWEVER, if you were standing on one of the poles, you would not be rotating at
all. Therefore, weÕve gone awry in our logic, or need clarification somewhere.
Checking the validity of our statements:
/fontfamily>IF:/fontfamily>
A person standing on the pole is not rotating at all.
/fontfamily>AND:/fontfamily>
A person standing on the equator is rotating at 1,047 miles per hour.
/fontfamily>THEN:/fontfamily>
We need a measurement that accounts for this displacement that gives rise to
varying rotational speeds on Earth.
Clarifying our prior logic to include ALL cases:
/fontfamily>IF:/fontfamily>
I map out the movement of a general point on Earth.
/fontfamily>AND:/fontfamily>
I drop that movement pattern down to the equator.
/fontfamily>THEN:/fontfamily>
I will have the segment of the equator relevant to that general point.
/fontfamily>IF:/fontfamily>
I have the segment of the equator relevant to that general point.
/fontfamily>AND:/fontfamily>
I know the radius of spherical Earth is 4,000 miles.
/fontfamily>THEN:/fontfamily>
The general-point (radius) is found by: cos(x) =
general-radius/4000 miles
/fontfamily>IF:/fontfamily>
The general-point (radius) is found by: cos(x) =
general-radius/4000 miles
/fontfamily>THEN:/fontfamily>
The general radius = (4000 miles)(cos(x))
LetÕs check the two extremes:
/fontfamily>IF:/fontfamily>
The general radius = (4000 miles)(cos(x))
/fontfamily>AND:/fontfamily>
Angle (x) at the pole is 90 degrees.
/fontfamily>THEN:/fontfamily>
The radius at the pole is: Radius = (4000 miles)(cos90)=0
miles
Also:
/fontfamily>IF: /fontfamily>The
general radius = (4000 miles)(cos(x))
/fontfamily>AND:/fontfamily>
Angle (x) at the equator is 0 degrees.
/fontfamily>THEN:/fontfamily>
The radius at the equator is: Radius = (4000 miles)(cos0)=4000 miles
/fontfamily>/flushboth>Coordinate
Geometry on a Sphere
/fontfamily>/center>/smaller>/smaller>/fontfamily>IF:
/fontfamily>This angle maps out
circumference lines parallel to the equator called /fontfamily>LATITUDE
LINES/fontfamily> with 0 degrees the
Equator.
/fontfamily>AND:/fontfamily>
There must be a corresponding concept for lines /fontfamily>PERPENDICULAR/fontfamily>
to the equator. These are /fontfamily>LONGITUDE
LINES/fontfamily>, with 0 degrees =
/fontfamily>THEN:/fontfamily>
We have a Òcoordinate system,Ó similar to the Cartesian coordinate system, with
latitude baseline = equator, and longitude baseline =
And we can finally clean up our logic regarding the speed of rotation at any
point on earthÉ
/fontfamily>IF:/fontfamily>
The radius of circumference at a point Y on earth = 4000 miles x cos(latitude(Y).
/fontfamily>AND:/fontfamily>
The Earth makes one complete rotation every day.
/fontfamily>THEN:/fontfamily>
The rotational speed for a point Y on Earth =
2/fontfamily>¹/fontfamily>(4000m)cos(latitude (Y)
/fontfamily>24 hours
/fontfamily>AND:/fontfamily>
/fontfamily>THEN:/fontfamily>
What do we know about the /fontfamily>DIRECTION/fontfamily>
of this rotation?
/fontfamily>IF: /fontfamily>If
it is
/fontfamily>AND: /fontfamily>Time
differences are a barometer of daylight.
/fontfamily>THEN: /fontfamily>ItÕs
daylight earlier in
/fontfamily>IF: /fontfamily>ItÕs
daylight earlier in
/fontfamily>AND IF:
/fontfamily>THEN: /fontfamily>Sunlight
strikes to the east, and moves westward.
/fontfamily>IF: /fontfamily>Sunlight
strikes to the east, and moves westward.
/fontfamily>THEN: /fontfamily>The
rotation of the earth is /fontfamily>COUNTERCLOCKWISE/fontfamily>.
And what about the Shape of the /fontfamily>ORBIT/fontfamily>
of the Earth as it /fontfamily>REVOLVES/fontfamily>
about the Sun? How elliptical/orbital is this revolutionary shape? And what is
the speed of this revolution?
/fontfamily>/flushboth>Circular?
Elliptical?
/fontfamily>/center>/smaller>/smaller>/fontfamily>IF:/fontfamily>
A circle has eccentricity = 0
/fontfamily>/flushboth>AND: /fontfamily>A
flattened circle has eccentricity near 1.
/fontfamily>THEN:/fontfamily>
We might guess, given all the drawings we see of the earthÕs orbit, the earth
has an orbital eccentricity = .5
/fontfamily>IF:/fontfamily>
The earth actually has an eccentricity = 0.0167
/fontfamily>THEN:/fontfamily>
The earthÕs orbit is almost CIRCULAR, and not elliptical.
/fontfamily>AND:/fontfamily>
The data is as follows: Minimum distance = 91m miles, the Maximum distance =
95m miles, and the Mean distance = 93m miles
/fontfamily>THEN:/fontfamily>
Then the maximum and minimum distance of the earth from the sun should be
nearly equal.
/fontfamily>Further
Notes from the Conference:
/fontfamily>Questions arising in the
presentation: how would you address the following reservations (challenges to
the clarity or causality of a statement):
¥ The earth is not an exact sphere, nor is the radius exactly 4000 miles. Would
you introduce significant digits in using this initial estimation in further
calculations?
¥ One day does not equal exactly 24 hours, but rather, 23 hours, 56 minutes, 4
seconds. How would you clarify this AND account for both figures?
¥ Rotational perspective: how would you account for the Òcounter-clockwiseÓ
challenge; that is, such a description depends on the perspective of the
viewer?
¥ How would you integrate the hour issue above with calendars, leap years, and
the need to keep daylight and seasonality Òin syncÓ with our expectations of
reality?
The Theory of Constraints for Education: Narrative Summary:
By using structured and visual thinking processÑclarity, causality,
predicted effect, additional cause, etc.Ñnot only are we able to arrive at a
genuine understanding of the subject, but in a manner consistent with ÒJoy in
WorkÓ. Further, one quickly realizes the majority of ÒfactsÓ in the universe
need not be Òmemorized for tests,Ó but are there for our discovery!
EDITORSÕ NOTES
/bigger>/fontfamily>(2)/fontfamily>
Kay Buckner-Seal, Cheryl A. Edwards
/fontfamily>We would like to thank Mike
Round for sharing this novel application of the thinking tools. You may view
this information as Mike intended in a PowerPoint presentation, which is
attached. Please note that the pdf version of TACTics is also attached. You must have Acrobat Reader to
open the file. It is freely available for download from:
/fontfamily>/flushboth>www.adobe.com/products/acrobat/readstep.html/color>.
/fontfamily>/center>If
you have the Reader installed but still can't open the file, drag it from this
e-mail to your desktop, launch the Reader, and open from the FILE menu. You may
also view TACTics in its intended formatting, by
visiting our web site at www.tocforeducation.com/color>.
Click on ÒWhatÕs New.Ó
/smaller>/smaller>We would love to hear from you. We
look forward to publishing any feedback that you may have about any of our
articles, your applications of the thinking processes, lessons, announcements,
and etc. You may send them by mail to: Cheryl A. Edwards,