Date: Oct 3, 2012 1:25 PM
Author: kirby urner
Subject: Re: An Algebra 2 Test

On Wed, Oct 3, 2012 at 9:29 AM, Dave L. Renfro <renfr1dl@cmich.edu> wrote:
> Robert Hansen wrote:
>
> http://mathforum.org/kb/message.jspa?messageID=7899711
>

>> Given the list I posted as a start (which I repeat below
>> along with a few topics I omitted)...
>>
>> Polynomial Arithmetic
>> Factoring and Simplification
>> Simultaneous Equations
>> Graphing
>> Probability
>> Conic Sections
>> Binomial Expansion
>> Logs and Exponents
>> Absolute Value and Radicals
>> Complex Numbers
>> Series
>> Composite Functions
>> Combinatorics
>>
>> What would you limit this list to? This is almost identical
>> to the Dolciani S&M book minus the trig.

>
> Perhaps the schools in my area were slow, as I don't
> believe any of them got much past about halfway through
> the Dolciani Algebra 2 book -- no trig., no logarithms,
> no conic sections, no probability, no combinatorics, and
> no binomial expansions at the Pascal's triangle level or
> above. And this was with perhaps at most 25% of the student
> body taking Algebra 2 at some point in high school. All the
> other topics from among those you listed, and some more such
> as mathematical induction and an introduction to calculus,
> were covered in the 4th year math course (variously called,
> depending on the school, "precalculus" or "mathematical
> analysis").


Not expanding Pascal's Triangle is kinda sad. I have it as one of
four Focal Points in my "focal points" blog post. [1]

Generating series and sequences according to rules, very important.

But more important is developing the left-right bridge between lexical
and graphical content, algebraic and geometric.

Pascal's Triangle contains the triangular and tetrahedral numbers,
envisioned as stacks of balls ala 'The Book of Numbers' by Conway and
Guy.

Bridging lexical and graphical analytically using XYZ apparatus
(coordinate systems) is not the place to begin.

So-called "gnomon studies" are more primitive (see 'Gnomon' by
Gazale). Figurate numbers, polyhedral numbers...

1, 12, 42, 92, 162... is another important series, having to do with
the number of balls in successive shells of a growing cuboctahedron.

This series will not be harped on by most 1900s texts as the geometry
of geodesic spheres, fullerene, virus shells, micro-organisms is not
topical, side-bar treatment at best.

Not telling stories about architecture, interweaving with the
mathematics, is another one of the gross deficiencies that deprives
students of their heritage.

>
> But that was then and there. For now, I would drop conic
> sections, maybe just touch on logarithms (focus mostly on
> solving equations such as 2^(x+3) = 5 and some very basic
> rewriting of expressions involving logarithms, and leave
> for precalculus things like logarithm graphs and the base
> change formula), perhaps do binomial expansions by relating
> it to combinatorics (e.g. when expanding (x + y)^8, the number
> of x^3*y^5 terms will be the number of times 3 x's and 5 y's
> appear when you list all 8-term ordered sequences ########
> in which each # is an x or a y), perhaps restrict radical
> equations to those that can be solved by squaring both sides
> at most once, skip composite functions (I've always felt
> that precalculus is where you make the transition from
> numbers and operations on numbers to functions and operations
> on functions), and only deal with complex numbers to the extent
> of adding, subtracting, multiplying, dividing, and graphing
> them and using them in solving quadratic equations.
>


When it comes to sequences, it's important to categorize them as:
a) divergent
b) convergent
c) oscillatory
d) chaotic

Oscillatory includes any repeating pattern, as when adding
sine waves of different amplitude and period ala Fourier.

Convergence takes us to concepts of limit and limiting value.

Chaotic would include series that feed back into themselves
and show great sensitivity to initial conditions.

In STEM, there's an emphasis on phasing in physics. Over on the PER
list we've been dissecting the Eureka! series from TVOntario some.[2]

> I think including some material on sequences and series is
> fine, although in my opinion many books are way overly formula
> oriented when dealing with arithmetic and geometric sequences
> and series. For example, instead of using a formula to
> determine how many terms are in the arithmetic sequence
> 16, 23, 30, ..., 681, notice that subtracting 16 from each
> of the terms doesn't change how many terms there are,
> which gives 0, 7, 14, ..., 665. Now notice that dividing
> each term by 7 doesn't change how many terms there are,
> which gives 0, 1, 2, ..., 95. Finally, notice that adding
> 1 to each term doesn't change how many terms there are,
> which gives 1, 2, 3, ..., 96. At this point it's obvious
> there are 96 terms, since if you start counting them, the
> count of the term is equal to the term. (Yes, this is like
> your "street fighting math" method you posted at [1].)
>


I would replace all this with revamped segments more like those discussed above.

1, 12, 42, 92, 162... are also the "icosahedral numbers" because of
how a cuboctahedron may transform into an icosahedron without loss of
"balls" -- the so-called Jitterbug Transformation in the
literature.[3]

> You can also incorporate some elementary algebra in
> obtaining various sequences and series results, as illustrated
> in the handout "seq-and-series.pdf" that I posted at [2].
>
> [1] http://mathforum.org/kb/message.jspa?messageID=7873200
>
> [2] http://mathforum.org/kb/message.jspa?messageID=6544585
>
> Dave L. Renfro


geom-growth.pdf (106.2 K)

...would be a place to branch off. 1, 12, 42, 92... corresponds to a
cumulative series 1, 13, 55, 147... which is an expanding FCC lattice
of equi-distant unit-radius points, an ideal gas model. How many
layers (what frequency?) would be needed before the cumulative number
reached Avogadro's Number? That's a typical STEM type question. [4]

Kirby

[1] http://mybizmo.blogspot.com/2006/09/focal-points.html (Focal Points)

[2] http://www.youtube.com/playlist?list=PLE82493DFAB9EA6FD
(Eureka! TV Ontario. Play list)

[3] http://4dsolutions.net/ocn/numeracy0.html (early algebra /
geometry for STEM teachers)

[4] http://controlroom.blogspot.com/2009/06/avogadros-number.html
(using FCC as ideal gas model -- average positions, uniform density
distribution)