[Edu-sig] Moving ahead in Alaska (long)

[kirby urner]

Did anyone else notice the Google valentines day graphic could be
construed as XO promotion?  I'm going to do something fun in my
blog about that.

Here's something long, but maybe useful if you're a math teacher
using Python for some reason (still an elite and rare breed -- looks
good on your resume):

On Fri, Feb 13, 2009 at 12:02 PM, kirby urner <kirby.urner at gmail.com> wrote:
<< SNIP >>

> Recent meeting with Anna Roys, TECC/Alaska (tecc-alaska.org):
>
> Lesson plan:  On-line Dictionary of Integer Sequences, enter 1, 12, 42, 92...
>
> Follow some links, to my page included, even if just for the pictures
> (good Virus from Life -- made out of metal nuts it looks like).
> Treasure hunt?

==== < INTERJECTION > =====

Here's something I just uploaded to Math Forum on this lovely Valentine's Day,
giving more context and background re my curriculum work with Alaska, the
Matsu District in particular.  It doesn't mention Python per se, but does talk
about computer programming.  Scroll to the very bottom for the rest of the
edu-sig context, as Python generators of the "must have" variety are part
of the mix...

> Kirby
>
> [ in a meeting with Anna Roys of TECC/Alaska, Kennedy
> School (McMenamins network), usa.pdx.or ]
>
> ------- End of Forwarded Message

Happy V-day ya'll!

Regarding the above meeting, I'm still putting puzzle
pieces together, will be for awhile, but I wanted to go
over a "signature lesson plan" that we use when suggesting
a charter distinguish itself from the pack.

Of course "distinguishing oneself from the pack" is not
always what a school wants to do, at least not on the
basis of curriculum. In sports it's OK (e.g. "best
football team") but in academics you're looking for
uniformity at the district level a lot of the time, so
that parents don't get all bent out of shape about one
school being oh so much better than another. A
traditional district may well aim for homogeneity, as
may a traditional state or even nation.

Actually, the more accurate description is that the
standards-based teaching methods now widely adopted aim
to put a "floor" under a given school, such that content
above and beyond what's in the standards is not actively
discouraged so much as put on a back burner next to
what's widely agreed as important. To use the football
analogy: get as good as you like, but at least make sure
your team plays by the rules, knows what these are.

So, back to the lesson plan, which we're able to hook to
standards here and there...

Consider the figurate numbers, such as the triangular and
square numbers. Review some example (previous lessons).
Use ping pong balls or other balls (clay OK), to model
the figures. Dots on paper also work for one layer, i.e.
for arrays in a plane.

Today we're ready to tackle some "polyhedral numbers"
meaning we come off the plane, start stacking layers.
Two easy examples, based on our work with the square and
triangular numbers: the half-octahedral and tetrahedral
numbers. Just stack consecutive layers from the previous
review starting with 1. This is where having actual
balls, not just dots on paper, might be advantageous.

Square: 1, 4, 9, 16...
Triangular: 1, 3, 6, 10...

Half-octahedral: 1, 1+4 = 5, 1+4+9 = 14, 1+4+9+16 = 30
Tetrahedral: 1, 1+3=4, 1+3+6 = 10, 1+3+6+10 = 20

A next step requires the Internet, although pre-printed
handouts would be another possibility. Visit the
On-Line Encyclopedia of Interger Sequences and enter
the above sequences, just to confirm they're in there and
to take a gander at some of the literature. A purpose
here is to connect with the larger culture and get a
sense of the knowledge base, its "humongousness" if you
will.

The "half-octahedral" sequence is actually called the
"square pyramidal" sequence (same thing):

http://www.research.att.com/~njas/sequences/A000330

And here are the tetrahedral numbers:
http://www.research.att.com/~njas/sequences/A000292

Lots of literature, plenty of branch points to other
topics.

What we're looking for here is a bridge to the sciences.
We'll find many of course, as both the square pyramidal
and tetrahedral ball packings define the all important
FCC lattice (also known as the CCP -- or even IVM in more
esoteric writings). Alexander Graham Bell's engineering
around towers and kits forms an architectural connection.
Other examples of this space frame or truss are not
difficult to find (here in Portland we have some great
examples).[1]

However, there's another way of getting to the FCC that
starts with a single nuclear ball with 12 packed around
it, all intertangent to one another. 12 equi-diameter
balls will squeeze around a nuclear one in more than one
way, but if you insist on maximizing points of
intertangency, then you're basically looking at two ways,
defined as the CCP and HCP respectively.

So how much of the above crystallographic background goes
into the lesson, and how much is for teacher notes? In
this overview of the plan, I don't make those decisions.
A lot depends on grade level and which standards we're
meeting.

In taking the CCP route, starting with 12-around-1, we're
able to expand outward with successive layers. The shape
is cuboctahedral, meaning we'll have probably needed some
background in the various polyhedra, at least the most
commonly encountered simple ones, such as the Platonic
Five and a smattering of others. Since we're doing
sphere packing, the rhombic dodecahedron (a zonohedron)
is going to be quite important (links to Kepler). So yes,
other lesson plans are implied here.

To make a long story short, the number of balls in the
successive layers of our growing cuboctahedron (1, 12,
42, 92...) is also the number of balls in successive
icosahedral arrangements. A great way to get this across
is to show how a cuboctahedron may transform into an
icosahedron and vice versa. Animations enter the picture
at this point, or many math classrooms will be equipped
with a sticks-and-rubber-joins affair such as we see in
this YouTube: http://www.youtube.com/watch?v=HefLC3PW8XQ

Back to the Encyclopedia:
http://www.research.att.com/~njas/sequences/A005901

You'll see in the header that "cuboctahedral" and
"icosahedral" are both mentioned.

The interesting fact about the icosahedral numbers is
they lay the groundwork for talking about (a) geodesic
spheres, wherein the vertices (balls) are perhaps
equidistant from some center and (b) the structure of
the virus, consisting of proteins called capsomeres that
follow the above sequence.[2] So this is a bridge from
mathematics to both architecture and naturally occurring
micro-architecture. The virus is our bridge to talking
about RNA-DNA in other lessons, when we've turned to
other subjects in biology etc.

For those charters working a computer language into their
curriculum (a growing number in some districts), these
are typically some of the shortest "programs" one might
wish for. However, jumping ahead to the closed form
algebraic expressions which generate the above will often
be done in conjunction with specific proofs. Mathematical
induction is one option, though in the case of the
cuboctahedral numbers in particular, I favor a different
approach, likewise algebraic.[3]

I'm expecting you'll discover these interconnected topics
making a lot of headway in some of the charters, as we
have corporate sponsors lining up to develop their market
potential and lots of green lights from university
departments. Alaska Pacific University and the State
University of Alaska work with the State of Alaska on the
development of standards (the former in particular) and
so are aware of where the connect points might be. TECC
itself is looking at the computer language piece, an
anticipation of standing out as a flagship in the Matsu
District. Advertising oneself as "early college" (in the
sense of encouraging cross-enrollment) and STEM (into
science and math) somewhat requires backing that up with
at least some computer programming activities. The days
when just calculators were sufficient appear to be behind
us where "technology in the classroom" is concerned. Both
the UK and US are moving to embrace low cost open source
tools (hence the coin "gnu math").

What I'm suggesting to subscribers to math-teach is they
mine this same network of interconnected topics if wanting
to differentiate above the "floor level" (bare minimum).
The STEM aspects are obvious, connections to contemporary
engineering and science manifest, opportunities for
computer use clear. Yes, Alaska may be ahead of the pack
at this point, but the lower 48 have opportunities to
borrow, especially if not in lockstep based on some text
book series that excludes all of the above. Those would
be the "non-competitive" schools, many of them mired in
outmoded curricula. If you're a parent, you now know
what to put on your radar, if looking for signs of
innovation and a commitment to STEM.

Kirby

[1] http://www.portlandbridges.com/00,D300CRW05758,24,0,1,0-portland-oregon.html

[2] http://books.google.com/books?id=7rfpzW7eMW4C&pg=PA181&lpg=PA181&dq=capsomeres+1,+12,+42&source=web&ots=3E4rWZjam3&sig=zRtjB9Lui5ncH-UxmFB3b9oSQ0A&hl=en&ei=iACXSdqkKZKasAOI9YyFAQ&sa=X&oi=book_result&resnum=2&ct=result

[3] http://mybizmo.blogspot.com/2007/01/gnu-math-memo.html

==== < /INTERJECTION > =====

So the functions below need to be rewritten as generators, mixed in with
our Fibonacci generator, Pascal's triangle generator, plus the prime and
chaotic sequence generators we've been discussing (acknowledging that
the "generator" may not always be the most computationally efficient
method -- sometimes efficiency takes a back seat  for pedagogical
purposes).

>
> We're focused on linking algebraic sequences, generator type stuff, to
> visual imagery, imaginary content, like we do later with coordinate
> systems (XYZ, spherical...), but "figurate numbers" ("polyhedral
> numbers") are a first bridge between algebra and geometry, coordinates
> be damned (until later).
>
> Glue four ping pong balls together:  voila, a tetrahedron (your unit
> of volume in some curriculum segments, unless your school is some kind
> of joke -- Alaska leading the pack here in some ways).
>

<< SNIP >>

> A quick challenge:
>
> Spheres packing around a nuclear sphere go 1, 12, 42, 92... 10*L*L +
> 2, where L is the layer number, except where L = 1 we have just the
> one ball (the shape is a cuboctahedron).  So how many balls total?
> Add up all the layers.  Yes, very easy to do in APL.
>
> In Python:
>
> def cubocta( layer ):
>    if layer == 1:  return 1
>    return 10 * layer ** 2 + 2
>
> def total_balls( layer ):
>    total = 0
>    for i in range(1, layer + 1):
>        total = total + cubocta( i )
>    return total
>
> But isn't there a closed form algebraic expression for total_balls
> that doesn't require cumulative adding?  Damn straight.  We'll get to
> it.
>
> Don't forget to watch the cartoons!  This isn't Bourbaki.
> Visualizations encouraged!  This is MVC.