
Re: Alternative area postulate for geometry
Posted:
Jul 5, 2012 2:18 PM


On Thu, Jul 5, 2012 at 5:37 AM, Robert Hansen <bob@rsccore.com> wrote: > The most common way of teaching multiplication (that I have seen) is with groups of objects and MxN arrays (grids) of objects. I think it works very well (like the number line does in addition) but not without good (and repeated) exposition like your example. This is why I am so fond of arithmetic at these ages. It connects you to "number" and makes all this exposition meaningful and lasting. > > Bob Hansen >
In intelligent approach to area includes using the figurate numbers, as this works with volume also, where number sequences become associated with polyhedrons.
This approach is exploited quite gracefully in 'The Book of Numbers' by Conway and Guy, a charming little tome.
The triangular numbers are so simple because each row of the triangle has one more ball in it: 1, 2, 3, 4... with the totals in each triangle being successive cumulative totals: 1, 3, 6, 10....
There's a sweet visual proof that two consecutive triangular numbers make a square number, e.g. 3+6=9, 6+10=16 and so on.
However these "square numbers" may be organized as a sequence of equilateral triangles as well, per the template below:
http://www.flickr.com/photos/17157315@N00/7509355378/in/set72157630437640906
Having this information in poster form, either made by students themselves or imported, is another hallmark of a STEMready classroom.
Here's another such figure from a published book:
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
> > On Jul 4, 2012, at 11:13 PM, Joe Niederberger <niederberger@comcast.net> wrote: > >> Perhaps this kind of exposition would work better with real objects at first  a bottle of Coke: then two bottles of Coke.
Here's a picture of the "mixing bowl kit" I shared with some of the school systems, a canonical set of polyhedrons designed for pouring.
http://www.flickr.com/photos/17157315@N00/479693617/
http://www.flickr.com/photos/17157315@N00/4222223792/in/photostream/
The great thing about sequences like 1, 12, 42, 92.... which corresponds to omnidirectional ball packing, squeezing 12around1, then 42, then 92.... is they're so easy to program in one or two lines of code, in whatever language.
The OnLine Encyclopedia of Integer Sequences becomes a primary source in such work. Links to STEM topics everywhere, e.g. morphology of the virus.
http://oeis.org/A005901
This give students immediate access to interesting functions and a doorway to algebra, coming from geometric shapes.
And yet no coordinate system apparatus need be used at this point.
What we have here is a lexical  graphical bridge suitable for early STEM learning  a topic in my presentation in Vilnius (Europython), with 'The Book of Numbers' an important influence.
Sphere packing to make shapes (e.g. triangles), conjoined to number sequences (1, 3, 6, 10...), is a node / grand central I signify using Pascal's Triangle.
Savvy readers know how much branches off from here, from probability to Pascal's Tetrahedron.
Kirby

