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: