2D and 3D
Coordinate Geometry: Bottomup learning using ''Autograph''
Douglas Butler
debutler@argonet.co.uk
iCT Training Centre
Oundle School
UK
Abstract
Learning
Mathematics can be like a house of cards  it can all be wasted if there is
no solid foundation. There is growing evidence that the imaginative use dynamic
software such as ?utograph?can help students firm up these foundations
using a stepbystep approach from the bottom up. For example:
Quadratics [progressing from y=1, y=1+x, y=x? y=1+x+x?to y=ax?bx+c
and the factorized form]; the link between completing the square and the vector
translation of a quadratic; forming a quadratic from three points, ... etc.
Differential equations [progressing from y'=1, y'=x, y'=y, to the implicit forms:
y'y=0, y'y=1, y'y=x, ... etc. Likewise 2nd order from y?0 to the general
damped second order].
The transition from 2D to 3D coordinate geometry can also be made easier by
visualising the 2D principles first: eg the vector [a,b] perpendicular to the
line ax+by=c, followed by the vector [a,b,c] perpendicular to the plane ax+by+cz=d.
Likewise, a firm understanding of conics in 2D will make 3D equations such as
x?y?z [paraboloid] easier to understand.
Similar
strategies are available in statistics using simple data sets to create histograms,
cumulative frequency curves and sampling distributions.
All the these ideas will be available as TurboDemo tutorials on the web for
this conference, reestablishing the principle "Author once, learn anywhere!"
, at www.tsmresources.com
