Date: Oct 3, 2012 12:29 PM Author: Dave L. Renfro Subject: Re: An Algebra 2 Test 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").

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.

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].)

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