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 <[email protected]> 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. _______________________________________________ Edu-sig mailing list [email protected] http://mail.python.org/mailman/listinfo/edu-sig
