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## 2D and 3D Coordinate Geometry: Bottom-up 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 step-by-step 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, re-establishing the principle "Author once, learn anywhere!" , at www.tsm-resources.com