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

Robert Hansen wrote:

> 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

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].



Dave L. Renfro