High school algebra 1 topics - have they changed recently?

[jasonRF]

Hello,

One of my children is switching to a new school system next year, and I was surprised to find that they cover subjects such as fractional exponents, exponential functions, rational functions and perhaps other topics as part of algebra 1. Their book has "common core" in the title, if that matters. It appears they revisit those same topics (hopefully in more depth) in algebra 2. I don't recall covering that material at all until algebra 2, but then I took algebra 30+ years ago. Do any of you teachers out there have experience with a similar syllabus, and does your experience indicate that it is better or worse than an algebra 1 course that focuses on the basics? By basics I am thinking linear equations, inequalities, systems of equations and inequalities, integer exponents, polynomials and factoring, and quadratic equations and applications. That is, what I recall learning in algebra 1.

Thanks!

Jason

 

[symbolipoint]

jasonF

Hard to say. About 15 years ago, Algebra 1 was still Algebra 1 the way you knew it; and Algebra 2 was Algebra 2 the way you knew it. I have nothing to say about "common core". The equivalent college courses were the same way, just done in shorter time. Something was not right when districts switched to textbooks which supposedly "met State standards". Some of the books seem/ed to become deficient. My opinion is that textbooks of Algebra 1 & 2 from 30 to 45 years ago were very good.

 

[MidgetDwarf]

We did exponents, exponential functions, and rational functions in Algebra 2. I am 27 years of age. I work with children, and I noticed that the common core books, appear to be more akin to comic books, then actual math books. Some of the problems are really interesting. However, it requires the student to have a good teacher that supplies the explanation and background. There really is no explanation or background in the common core books. The student is at the mercy of a good teacher, and their not that many "good" math teachers in k-12 in the US.

I would advise to go over a traditional math book "for fun," on the side with your child.

 

[mathwonk]

i recommend getting hold of a copy of Harold Jacobson's elememntary algebra book, iof you want your child to actually understand algebra,n regardless of the list of topics they "cover" in your book.

https://www.amazon.com/gp/product/0716710471/?tag=pfamazon01-20

by the way, other than understanding the meaning of the symbols, the key to understanding algebra is grasping the concept of a ":variable", and the most important theorems in basic algebra are the division algorithm for polynomials, the consequent root/factor theorem, and the rational root theorem.

the quadratic formula is nice but seldom taught today in a way that explains what is going on. The ancient books made that clear. i.e. in a quadratic equation X^2-bX +c = 0, the coefficient b is the sum r+s = b of the roots r,s and the coefficient c is the product c = rs of those roots. Hence if we could learn the difference of the roots r-s, then we could combine that with b and get the roots themselves. but fortunately the sum and the difference have similar squares, i.e. they only differ by 4 times the product, i.e. ((r-s)^2 +4rs = (r+s)^2 = b^2. hence (r-s)^2 = b^2 - 4rs = b^2 -4c. Thus r and s can be solved for by 2r, 2s = (r+s) ± (r-s) = b ± sqrt(b^2-4c).

this is less important then the previously mentioned items.

fractional exponents are a trivial consequence of the basic addition rules for exponents. i.e.since a^r.a^s = a^(r+s), necessarily a^(1/2).a^(1/2) = a^(1/2+1/2) = a^1 = a, so a^1/2 = sqrt(a). thus learning the basic rules are more important than memorizing these special facts.

perhaps the greatest elementary - advanced algebra book is that by euler, elements of algebra. in that work he treats fractional exponents in the first section, after the basic properties of arithmetic operations and powers.

here is an appreciation of it:

https://plus.maths.org/content/tale-two-curricula-eulers-algebra-text-book

euler explains solving even cubic equations so clearly that afterwards i was able to explain it to (brilliant) 10 year olds.

 

[symbolipoint]

mathwonk said this :

...by the way, other than understanding the meaning of the symbols, the key to understanding algebra is grasping the concept of a ":variable", and the most important theorems in basic algebra are the division algorithm for polynomials, the consequent root/factor theorem, and the rational root theorem.
...

The concept of numeric variable was never trouble for me. It WAS for other students, trouble for them. I often wondered and still do, why this was a difficult concept for so many students.

The only difference in any courses that STILL SHOULD BE, is the distinctions between Algebra 1, Algebra 2, and "College Algebra". Factor Theorem and Remainder Theorem, from what I clearly remember, were part of College Algebra but not Intermediate Algebra. We still did deal with roots of some polynomials, especially quadratics. Intermediate level did intensely teach general quadratic formula solution and also Conic Sections (but seemed to be only for two-dimensions).

 

[MidgetDwarf]

mathwonk said this :


The concept of numeric variable was never trouble for me. It WAS for other students, trouble for them. I often wondered and still do, why this was a difficult concept for so many students.

The only difference in any courses that STILL SHOULD BE, is the distinctions between Algebra 1, Algebra 2, and "College Algebra". Factor Theorem and Remainder Theorem, from what I clearly remember, were part of College Algebra but not Intermediate Algebra. We still did deal with roots of some polynomials, especially quadratics. Intermediate level did intensely teach general quadratic formula solution and also Conic Sections (but seemed to be only for two-dimensions).

Whats amazing is too, is that a lot of the community colleges in California are removing conic sections from the Intermediate algebra class. It is now called optional. Even more scary, is that it is being ignored in pre-calculus.

I know for a fact that this policy has been implemented in the los angeles community college district.
I fear for us all!

 

[symbolipoint]

Whats amazing is too, is that a lot of the community colleges in California are removing conic sections from the Intermediate algebra class. It is now called optional. Even more scary, is that it is being ignored in pre-calculus.

I know for a fact that this policy has been implemented in the los angeles community college district.
I fear for us all!

Bad! Really bad! To cut that out of Pre-Calculus/College Algebra? It is just too essential, and MUST be included in both. Quadratic Formula studies is a prerequisite to dealing with Conic Sections, and Conic Sections is also a typical and necessary part of studying Calculus, useful in Physics, Engineering, often which include some studies of Optics.

 

[Bystander]

The concept of numeric variable was never trouble for me. It WAS for other students, trouble for them. I often wondered and still do, why this was a difficult concept for so many students.

"x" is introduced as "the unknown" to be solved for in some courses, and that is the only definition of "x" that sticks in student minds, that it is an "unknown."

 

[jasonRF]

Thank you all for your comments. I may pick up a copy of Jacobson's book. Older books seem to be better for basic subjects anyway.

Cheers!

Jason