id: 06108984
dt: j
an: 2012f.00728
au: Herman, Marlena
ti: Exploring conics: why does $B^2 - 4AC$ matter?
so: Math. Teach. (Reston) 105, No. 7, 526-532 (2012).
py: 2012
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: G70
ut: equations; geometry; geometric concepts; mathematics instruction;
manipulative materials; computer software; educational technology;
conic sections
ci:
li: doi:10.5951/mathteacher.105.7.0526
http://www.nctm.org/publications/article.aspx?id=32203
ab: Summary: The ancient Greeks studied conic sections from a geometric point
of view ‒ by cutting a cone with a plane. Later, Apollonius (ca.
262‒190 BCE) obtained the conic sections from one right double cone.
The modern approach to the study of conics can be considered
“analytic geometry,” in which conic sections are defined in terms
of distance relationships or described as graphs of certain types of
equations. Distance relationships involve the center of the shape, the
focus or foci, the directrix, and the position of axes. Thus, the set
of all points equidistant from a fixed point (the “center") makes up
a circle; the set of all points in which the sum of the distances to
two fixed points (the “foci”) is constant makes up an ellipse; the
set of all points in which the difference of the distances to two fixed
points (the foci) is constant makes up a hyperbola; and the set of all
points in which the distance to a fixed point (focus) is equal to the
distance to a fixed line (the “directrix”) makes up a parabola. In
this article, the author describes how paper-folding activities using
waxed paper can help students explore geometric properties of the conic
sections. She also discusses how an introduction to definitions and
equations of conic sections can be extended to explain the significance
of the “discriminant.” (ERIC)
rv: