Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
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/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 Peter Duveen
6/25/12 Robert Hansen
6/25/12 Gary Tupper
6/25/12 Robert Hansen
6/25/12 Gary Tupper
6/25/12 Robert Hansen
6/25/12 kirby urner
6/25/12 Peter Duveen
6/25/12 kirby urner
6/26/12 Robert Hansen
6/26/12 Gary Tupper
6/28/12 Gary Tupper
6/28/12 Robert Hansen
6/29/12 Wayne Bishop
6/29/12 Gary Tupper
6/29/12 Wayne Bishop
6/26/12 Joe Niederberger
6/26/12 Joe Niederberger
6/26/12 Peter Duveen
6/26/12 Robert Hansen
6/27/12 Wayne Bishop
6/28/12 kirby urner
6/28/12 Wayne Bishop
6/28/12 kirby urner
6/26/12 Joe Niederberger
6/26/12 Peter Duveen
6/26/12 Peter Duveen
6/27/12 Robert Hansen
6/26/12 kirby urner
6/27/12 Wayne Bishop
6/27/12 Robert Hansen
6/27/12 Peter Duveen
6/27/12 Peter Duveen
6/27/12 Joe Niederberger
6/27/12 kirby urner
6/27/12 Joe Niederberger
6/27/12 Robert Hansen
6/27/12 Joe Niederberger
6/27/12 Joe Niederberger
6/27/12 Peter Duveen
6/27/12 Joe Niederberger
6/28/12 Peter Duveen
6/28/12 Joe Niederberger
6/28/12 Wayne Bishop
6/28/12 Joe Niederberger
6/28/12 Joe Niederberger
6/29/12 Wayne Bishop
6/29/12 Peter Duveen
6/29/12 Joe Niederberger
6/29/12 Robert Hansen
6/29/12 Wayne Bishop
6/30/12 kirby urner
6/30/12 Robert Hansen
6/29/12 Joe Niederberger
6/29/12 Robert Hansen
6/29/12 kirby urner
6/29/12 Wayne Bishop
6/29/12 Joe Niederberger
6/29/12 Robert Hansen
6/29/12 Robert Hansen
6/29/12 Joe Niederberger
6/29/12 Robert Hansen
6/29/12 Robert Hansen
6/29/12 Joe Niederberger
6/29/12 Joe Niederberger
6/29/12 Robert Hansen
6/30/12 Louis Talman
6/30/12 Robert Hansen
6/30/12 Wayne Bishop
7/1/12 Louis Talman
7/1/12 Louis Talman
7/1/12 Louis Talman
7/1/12 Wayne Bishop
6/30/12 Robert Hansen
7/1/12 Robert Hansen
7/1/12 Louis Talman
7/2/12 Robert Hansen
7/2/12 Wayne Bishop
7/3/12 Gary Tupper
7/3/12 Robert Hansen
7/2/12 kirby urner
7/2/12 Robert Hansen
7/2/12 kirby urner
7/2/12 Robert Hansen
7/2/12 kirby urner
7/2/12 Robert Hansen
7/2/12 Wayne Bishop
7/3/12 kirby urner
6/29/12 Peter Duveen
6/29/12 kirby urner
6/30/12 Wayne Bishop
6/30/12 kirby urner
6/30/12 Wayne Bishop
7/1/12 kirby urner
7/1/12 Wayne Bishop
7/2/12 kirby urner
7/2/12 Wayne Bishop
7/3/12 kirby urner
6/30/12 Joe Niederberger
6/30/12 Joe Niederberger
6/30/12 Peter Duveen
7/1/12 Robert Hansen
7/1/12 Wayne Bishop
7/1/12 Joe Niederberger
7/2/12 Peter Duveen
7/3/12 Joe Niederberger
7/3/12 Wayne Bishop
7/4/12 Wayne Bishop
7/4/12 Robert Hansen
7/4/12 kirby urner
7/3/12 Joe Niederberger
7/3/12 Joe Niederberger
7/18/12 kirby urner
7/4/12 Peter Duveen
7/4/12 kirby urner
7/4/12 Louis Talman
7/4/12 Robert Hansen
7/4/12 Louis Talman
7/4/12 Robert Hansen
7/4/12 Wayne Bishop
7/4/12 Peter Duveen
7/5/12 Robert Hansen
7/4/12 Joe Niederberger
7/5/12 kirby urner
7/5/12 Wayne Bishop
7/5/12 kirby urner
7/6/12 Robert Hansen
7/6/12 kirby urner
7/6/12 Wayne Bishop
7/6/12 Robert Hansen
7/6/12 kirby urner
7/4/12 Joe Niederberger
7/5/12 Robert Hansen
7/5/12 Wayne Bishop
7/5/12 kirby urner
7/5/12 Robert Hansen
7/5/12 kirby urner
7/4/12 Joe Niederberger
7/6/12 Joe Niederberger
7/6/12 kirby urner
7/6/12 Joe Niederberger
7/7/12 Wayne Bishop