id: 06560937
dt: j
an: 2016c.00767
au: Matsuura, Ryota; Sword, Sarah
ti: Illuminating coordinate geometry with algebraic symmetry.
so: Math. Teach. (Reston) 108, No. 6, 470-473 (2015).
py: 2015
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: G74 H24
ut: symmetric polynomials; coordinate geometry; analytic geometry; upper
secondary; triangles; proofs; centroid; center of mass; medians;
equations of straight lines; parametric equations; generalization;
tetrahedra; concurrency; reasoning
ci:
li: http://www.nctm.org/Publications/mathematics-teacher/2015/Vol108/Issue6/Illuminating-Coordinate-Geometry-with-Algebraic-Symmetry/
ab: From the text: A symmetric polynomial is a polynomial in one or more
variables in which swapping any pair of variables leaves the polynomial
unchanged. For example, $f(x,y,z)=xy+xz+yz$ is a symmetric polynomial.
If we interchange the variables $x$ and $y$, we obtain $yx+yz+xz$,
which is the same as $f(x,y,z)$; likewise, swapping $x$ and $z$ (or $y$
and $z$) returns the original polynomial. These polynomials arise in
many areas of mathematics, including Galois theory and combinatorics,
but they are rarely taught in a high school curriculum. In this
article, we describe an application of symmetric polynomials to a
familiar problem in coordinate geometry, thus introducing this powerful
tool in a context that is accessible to high school students.
rv: