Is “advanced algebra” a generally accepted term in HS education? Owing to the sequence I took, I am not precisely familiar with how mathematical concepts are partitioned in HS.
Here is the description of Advanced Algebra from my HS webpage.
Is this definition pretty consistent across schools?
Here is the College Board’s list of what will be covered in the New SAT math section. Among that list, what is considered “advanced algebra”?
There’s a pretty complete list on your link of what math concepts you’ll need for the SAT. You can also take a practice test.
But in high school? Well, my school (in California, if it matters) has Algebra 1-2 and Algebra 3-4, then also plus Pre-Calculus, which is essentially a review of Algebra 3-4, with a bit more detail on vectors, limits, and a few other things (at least, that’s my impression having taken the first semester of it). If someone asked about “the Advanced Algebra class,” I’d probably just say, “You mean the Algebra 3-4 class?”
Now, to fight some ignorance:
I vaguely recall a hardcore math doper saying that algebra is technically the study of polynomials and nothing else, but I think it’s generally accepted to include everything in that list. Am I remembering correctly?
basic set theory
introduction to the concepts of natural, rational, irrational, transcendental, prime, real, and complex numbers
basic matrix algebra
story problems
lines and planes
conic sections
trigonometry
polynomials
prime factors, LCM, GCD
logarithms and exponentials
scientific notation
quadratic equations
basic statistics
Eh. A high school algebra class is kind of a general introduction to math. Nothing on that list really falls under the heading of modern algebra, but it really isn’t meant to, so it’s all good. ccwaterback’s list is probably a pretty good guide to what you’ll find on the SAT, or at least to what you should be familiar with after three years of high school.
Just for the curious, modern algebra can be very succinctly described as the study of structure and actions that preserve it. Going into more detail than that would take some space, and is probably more than anyone wants.
In my highschool Pre-Calculus and Advanced Algebra covered the same material, but Pre-Calc covered it in a semester, and AA covered it in a year.
In related news, there was some math on the GRE “general” test that I had last reviewed in the 7th grade (a mere 9 years prior). I was thanking my lucky stars (and Mrs. Chan) as I pulled the ability to multiply polynomials out of my ass.
The description in the OP sounds like what is often referred to as “College Algebra” (it’s about what you’d find in a textbook titled College Algebra) or “Pre-Calculus” (except that that would uisually include trigonometry along with the algebra).
That’s about as advanced as algebra gets in the pre-calculus years (HS or early college), but to a mathematician, “algebra” refers to something much broader and more general than just the standard operations on real or complex numbers (as ultrafilter pointed out). If you’re not careful, you could pick up a book called Introduction to Algebra expecting to find “x + 3 = 7” and instead you get groups and rings and fields and Galois theory.
Nobody mentioned mathematical induction, deriving the binomial theorem, or theory of equations. Don’t they usually cover that in second year HS algebra?
Yes, Spectre, quite possibly. (Although I’m not sure what you mean by “theory of equations”; the OP did mention “roots of polynomial equations.”)
Of all the topics so far mentioned, some are sure to be covered, while others may or may not be depending on various factors.
I never saw induction until pre-calc, and I kinda got the impression that a lot of people in my college introductory discrete math class had never seen it before. And I had never seen a derivation of the binomial theorem until just now.
There were some interesting facts about polynomials covered in HS that were never proven, such as Descartes’ rule of signs. I never got far enough in field theory to see how it’s done.