TACTics Journal
A Publication for and by TOC for Education Practitioners
May 28, 2004


In this weekÕs issue:
Elementary/Secondary TACTics
(1) The Earth in the Solar System, Mike Round
EditorsÕ Notes
(2) Kay Buckner-Seal, Cheryl A. Edwards


ELEMENTARY/SECONDARY TACTICS
(1) The Earth in the Solar System: A Logical Look at Orbits, Seasons, Rotations and Revolutions
Mike Round, TOCFE Director, USA

LetÕs assume the distance from the EarthÕs core to the outer surface is 4,000 miles. What can we conclude from this?

IF: The distance from the EarthÕs core to the outer surface is 4,000 miles.
AND: The distance from the EarthÕs core to the outer surface is the RADIUS of the Earth.
THEN: The radius of the earth is 4,000 miles.

IF: The radius of the earth is 4,000 miles.
AND: The circumference of a sphere is equal to 2¹ (radius)
THEN: 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 SPEED of rotation?

IF: The Earth is a sphere.
AND: The distance from the EarthÕs core to the outer surface is 4,000 miles.
THEN: The circumference of the earth is equal to (2)(3.14)(4000 miles) = 25,120 miles.

IF: The circumference of the earth is equal to (2)(3.14)(4000 miles) = 25,120 miles.
AND: The Earth makes one complete rotation every day.
THEN: The Earth rotates at a speed of 25,120 miles / 24 hours.

IF: The Earth rotates at a speed of 25,120 miles / 24 hours.
THEN: 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.
THEN: 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:
IF: A person standing on the pole is not rotating at all.
AND: A person standing on the equator is rotating at 1,047 miles per hour.
THEN: 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:
IF: I map out the movement of a general point on Earth.
AND: I drop that movement pattern down to the equator.
THEN: I will have the segment of the equator relevant to that general point.

IF: I have the segment of the equator relevant to that general point.
AND: I know the radius of spherical Earth is 4,000 miles.
THEN: The general-point (radius) is found by: cos(x) = general-radius/4000 miles

IF: The general-point (radius) is found by: cos(x) = general-radius/4000 miles
THEN: The general radius = (4000 miles)(cos(x))

LetÕs check the two extremes:
IF: The general radius = (4000 miles)(cos(x))
AND: Angle (x) at the pole is 90 degrees.
THEN: The radius at the pole is: Radius = (4000 miles)(cos90)=0 miles

Also:
IF: The general radius = (4000 miles)(cos(x))
AND: Angle (x) at the equator is 0 degrees.
THEN: The radius at the equator is: Radius = (4000 miles)(cos0)=4000 miles


Coordinate Geometry on a Sphere

IF: This angle maps out circumference lines parallel to the equator called LATITUDE LINES with 0 degrees the Equator.
AND: There must be a corresponding concept for lines PERPENDICULAR to the equator. These are LONGITUDE LINES, with 0 degrees = Greenwich, England.
THEN: We have a Òcoordinate system,Ó similar to the Cartesian coordinate system, with latitude baseline = equator, and longitude baseline = Greenwich, England.

And we can finally clean up our logic regarding the speed of rotation at any point on earthÉ

IF: The radius of circumference at a point Y on earth = 4000 miles x cos(latitude(Y).
AND: The Earth makes one complete rotation every day.
THEN: The rotational speed for a point Y on Earth =
2¹(4000m)cos(latitude (Y)
24 hours
AND: Yakima is found at: 46 degrees North latitude, 120 degrees West longitude
THEN: Yakima rotates at a speed of 727 miles per hour.

What do we know about the DIRECTION of this rotation?

IF: If it is 10:00 AM in Kansas City, it is 8:00 AM in Yakima.
AND: Time differences are a barometer of daylight.
THEN: ItÕs daylight earlier in Kansas City than it is in Yakima.

IF: ItÕs daylight earlier in Kansas City than it is in Yakima.
AND IF: Kansas City is east of Yakima.
THEN: Sunlight strikes to the east, and moves westward.

IF: Sunlight strikes to the east, and moves westward.
THEN: The rotation of the earth is COUNTERCLOCKWISE.

And what about the Shape of the ORBIT of the Earth as it REVOLVES about the Sun? How elliptical/orbital is this revolutionary shape? And what is the speed of this revolution?


Circular? Elliptical?

IF: A circle has eccentricity = 0
AND: A flattened circle has eccentricity near 1.
THEN: We might guess, given all the drawings we see of the earthÕs orbit, the earth has an orbital eccentricity = .5

IF: The earth actually has an eccentricity = 0.0167
THEN: The earthÕs orbit is almost CIRCULAR, and not elliptical.
AND: The data is as follows: Minimum distance = 91m miles, the Maximum distance = 95m miles, and the Mean distance = 93m miles
THEN: Then the maximum and minimum distance of the earth from the sun should be nearly equal.

Further Notes from the Conference:
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
(2) Kay Buckner-Seal, Cheryl A. Edwards

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:
www.adobe.com/products/acrobat/readstep.html.
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. Click on ÒWhatÕs New.Ó

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, 2253 S. Hill Island Rd., Cedarville, Michigan 49719, USA. Or send hyperlink to cedwards@cedarville.net or kayseal@comcast.net.