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CONIC SECTIONSWriting activities in the classroom fall on a continuum from copying to free writing:
The learners can do the copying activities, guided writing by following the instructions and free writing such as poetic writing (from Scrivener, J.1994. Learning Teaching. Henemann. P. 157). Three-phase framework of teaching to write. The process of teaching to write includes “familiarization with similar pieces of writing”, “creation of written discourse” and “sharing pieces of writing in the group”. It is organized according to the three-phase framework: · Pre-writing (schemata activation, motivation for writing, preparation for the language, familiarization with the format of the target text) · While-writing (thesis development, writing from notes, ending up with a given phrase, proceeding from a given beginning phrase, following a plan, following a format and register, solving a problem) · Post-writing (reflection on the spelling and reasoning errors, sharing the writing with the group mates, redrafting, peer editing)
CONIC SECTIONS The ellipse and hyperbolaare known as central conics. Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible. A conic section may more formally be defined as the locus of a point A conic section with conic section directrix at
(Yates 1952, p. 36), where
for an ellipse,
for a parabola, and
for a hyperbola. The polar equation of a conic section with focal parameter
The pedal curve of a conic section with pedal point at a focus is either a circle or a line. In particular the ellipse pedal curve and hyperbola pedal curve are both circles, while the parabola pedal curve is a line (Hilbert and Cohn-Vossen 1999, pp. 25-27). Five points in a plane determine a conic (Coxeter and Greitzer 1967, p. 76; Le Lionnais 1983, p. 56; Wells 1991), as do five tangent lines in a plane (Wells 1991). This follows from the fact that a conic section is a quadratic curve, which has general form
so dividing through by
leaves five constants. Five points,
The general equation of a conic section in trilinear coordinates is
(Kimberling 1998, p. 234). For five points specified in trilinear coordinates
(Kimberling 1998, p. 235). Two conics that do not coincide or have an entire straight line in common cannot meet at more than four points (Hilbert and Cohn-Vossen 1999, pp. 24 and 160). There is an infinite family of conics touching four lines. However, of the eleven regions into which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses. Let a polygon of
Date: 2014-12-22; view: 2040
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