First, as far as I can tell, there are two "Big Ideas" of Algebra 1:

- Everything students already know about arithmetical expressions (adding, subtracting, multiplying, exponents, radicals, fractions, simplifying, word problems) can be extended in a consistent way to situations involving variables.
- Equations with variables give you the power to describe situations more generally than a single set of numbers allows.

I think one of the most problematic concepts of Algebra 1 is "Simplifying". It's so very easy to slip into "here are the rules for what constitutes simplified in this case, and this case, and this case..." (polynomials, exponents, rationals, radicals, ...) and so very difficult to access a big idea like, "We're trying to make our lives easier when we move on to the next step of solving or using a complicated equation that describes some situation in real life like particle physics or the stock market or the operation of an engine or the forces on a bridge." I can say that to students, but I have a hard time showing them or getting them to experience it...

Second, I find that I have some startlingly clear memories of my own Algebra 1 class in eighth grade in 1990, probably because some of the subject matter I learned never got used again. It seems like there's a lot of manipulation and variations on a theme that we teach students but aren't actually critical to their ongoing life as mathematical practicioners.

For example:

- I've used a lot of math in my life and graphed a lot of linear functions and data. All you really need is a deep and thorough understanding of y = mx + b. You do not need the point-slope form of a line. You might need the standard form when you get around to doing linear functions at a deeper level in Linear Algebra (and all of its applications in physics and economics and ...) in college, but could it wait until then?
- I've used a lot of math in my life, and while being able to factor out the greatest common factor of a polynomial is a deeply practical skill for all the math it can make easier, and factoring x^2 + bx + c offers a sort of aesthetic pleasure and can be used to ensure that students understand FOIL at a deeper, backwards operation level, no one ever tries to factor ax^2 + bx + c . You pull out the quadratic formula, because it's so much more general and so much less guess and check. Why do we then waste time teaching an essentially guess-and-check approach to solving it?