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Participants will study
topics in high school mathematics from a mature perspective.
Topics include Euclidean and non-Euclidean geometries, coordinate
geometry, conic sections, proof. Connections to classroom practice
will be included. Prerequisite: Math X467 (Perspectives on Geometry) and Math X464B (Perspectives on Functions 2)
EUCLIDEAN GEOMETRY (EUCLID)
Participants contrast the stipulative definitions common in mathematics
with the descriptive definitions that are common in everyday
life, and they discuss the importance of precise definitions
in mathematics. Participants learn about axiom systems through
an examination of Book 1 of Euclid’s Elements. Participants
find proofs of several of Euclid’s propositions, and they
present proofs in paragraph, two-column, and flowchart formats.
Participants explore different methods of teaching students the
necessity of axioms and how to organize and write up proofs.
SPHERICAL
GEOMETRY (SPHEREGEO)
Participants prove that the sum of the angles of a triangle in
a plane equals 180°. Participants discuss the definitions of
line and of triangle on a sphere. They use Lenart spheres to gather
data for the sum of the angles of a triangle on a sphere, and they
discuss their conjectures.
HYPERBOLIC GEOMETRY (HYPER)
Participants prove a theorem in Euclidean geometry by both analytic
and synthetic methods. They use it to help introduce a model
for the hyperbolic plane, and they explore the model by investigating
sums of angles for hyperbolic triangles. Participants see that
the parallel postulate does not hold in the hyperbolic plane,
and
they learn about the discovery of non-Euclidean geometry. Participants
compare and contrast the three geometries: Euclidean, spherical,
and hyperbolic.
CONIC SECTIONS: CIRCLES (CONIC1)
Participants use locus definitions to construct the circle using
string. They use coordinate geometry to find the equation of
a circle, and investigate other geometric relationships in
the coordinate
plane.
CONIC SECTIONS: PARABOLAS (CONIC2)
Participants use locus definitions to construct the parabola
by paper folding. They use relationships among the focus
and directrix
to find equations of parabolas. Applications are included.
CONIC
SECTIONS: ELLIPSES (CONIC3)
Participants use locus definition and string to construct
the ellipse. They use relationships among the major axis,
minor
axis, and foci
to determine equations. An application/project is included.
CONIC
SECTIONS: HYPERBOLAS (CONIC4)
Participants use locus definition to construct the hyperbola
by paper folding. They use relationships among the transverse
axis,
conjugate axis, and foci to determine equations. Participants
compare properties of conics, and they identify conics
from their general
form. Applications are included.
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