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Topic: Alternative area postulate for geometry
Replies: 141   Last Post: Jul 18, 2012 9:39 PM

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kirby urner

Posts: 2,022
Registered: 11/29/05
Re: Alternative area postulate for geometry
Posted: Jul 5, 2012 2:18 PM
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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/set-72157630437640906

Having this information in poster form, either made by students
themselves or imported, is another hallmark of a STEM-ready 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 omni-directional ball packing, squeezing 12-around-1,
then 42, then 92.... is they're so easy to program in one or two lines
of code, in whatever language.

The On-Line 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


Date Subject Author
6/25/12
Read Alternative area postulate for geometry
Peter Duveen
6/25/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/25/12
Read Re: Alternative area postulate for geometry
Gary Tupper
6/25/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/25/12
Read Re: Alternative area postulate for geometry
Gary Tupper
6/25/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/25/12
Read Re: Alternative area postulate for geometry
kirby urner
6/25/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/25/12
Read Re: Alternative area postulate for geometry
kirby urner
6/26/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/26/12
Read Re: Alternative area postulate for geometry
Gary Tupper
6/28/12
Read Re: Alternative area postulate for geometry
Gary Tupper
6/28/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/29/12
Read Re: Alternative area postulate for geometry
Gary Tupper
6/29/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/26/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/26/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/26/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/26/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/27/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/28/12
Read Re: Alternative area postulate for geometry
kirby urner
6/28/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/28/12
Read Re: Alternative area postulate for geometry
kirby urner
6/26/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/26/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/26/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/27/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/26/12
Read Re: Alternative area postulate for geometry
kirby urner
6/27/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/27/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/27/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/27/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/27/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/27/12
Read Re: Alternative area postulate for geometry
kirby urner
6/27/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/27/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/27/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/27/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/27/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/27/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/28/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/28/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/28/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/28/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/28/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/29/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/30/12
Read Re: Alternative area postulate for geometry
kirby urner
6/30/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
kirby urner
6/29/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/29/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/30/12
Read Re: Alternative area postulate for geometry
Louis Talman
6/30/12
Read Re: Alternative area postulate for geometry
Robert Hansen
6/30/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/1/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/1/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/1/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/1/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/30/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/1/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/1/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/2/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/2/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/3/12
Read Re: Alternative area postulate for geometry
Gary Tupper
7/3/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/2/12
Read Re: Alternative area postulate for geometry
kirby urner
7/2/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/2/12
Read Re: Alternative area postulate for geometry
kirby urner
7/2/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/2/12
Read Re: Alternative area postulate for geometry
kirby urner
7/2/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/2/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/3/12
Read Re: Alternative area postulate for geometry
kirby urner
6/29/12
Read Re: Alternative area postulate for geometry
Peter Duveen
6/29/12
Read Re: Alternative area postulate for geometry
kirby urner
6/30/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
6/30/12
Read Re: Alternative area postulate for geometry
kirby urner
6/30/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/1/12
Read Re: Alternative area postulate for geometry
kirby urner
7/1/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/2/12
Read Re: Alternative area postulate for geometry
kirby urner
7/2/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/3/12
Read Re: Alternative area postulate for geometry
kirby urner
6/30/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/30/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
6/30/12
Read Re: Alternative area postulate for geometry
Peter Duveen
7/1/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/1/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/1/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/2/12
Read Re: Alternative area postulate for geometry
Peter Duveen
7/3/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/3/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/4/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/4/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/4/12
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kirby urner
7/3/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/3/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/18/12
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kirby urner
7/4/12
Read Re: Alternative area postulate for geometry
Peter Duveen
7/4/12
Read Re: Alternative area postulate for geometry
kirby urner
7/4/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/4/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/4/12
Read Re: Alternative area postulate for geometry
Louis Talman
7/4/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/4/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/4/12
Read Re: Alternative area postulate for geometry
Peter Duveen
7/5/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/4/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/5/12
Read Re: Alternative area postulate for geometry
kirby urner
7/5/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/5/12
Read Re: Alternative area postulate for geometry
kirby urner
7/6/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/6/12
Read Re: Alternative area postulate for geometry
kirby urner
7/6/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/6/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/6/12
Read Re: Alternative area postulate for geometry
kirby urner
7/4/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/5/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/5/12
Read Re: Alternative area postulate for geometry
Wayne Bishop
7/5/12
Read Re: Alternative area postulate for geometry
kirby urner
7/5/12
Read Re: Alternative area postulate for geometry
Robert Hansen
7/5/12
Read Re: Alternative area postulate for geometry
kirby urner
7/4/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/6/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/6/12
Read Re: Alternative area postulate for geometry
kirby urner
7/6/12
Read Re: Alternative area postulate for geometry
Joe Niederberger
7/7/12
Read Re: Alternative area postulate for geometry
Wayne Bishop

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