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I am a pre-university student who wants to help students with Algebra 1 and 2 in high school. I am curious to how the curriculum was built and what the goal of teaching both algebra 1 and 2 might be. I am comfortable with Algebra, I just never understood why the two classes are structured the way they are. I would like to know this more (and review the curriculum of the two) and see how I can help with students taking the course. My math background is college algebra, trig, and calc 1 and I am fairly decent at solving algebraic equations.

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I don't think there is a single, central authority that defines a precise content standard of "Algebra I" and "Algebra II" in the US, but the OpenStax textbooks Elementary Algebra and Intermediate Algebra mostly align with my memory of the subjects. There seems to be a little variability in terms of the closing topic(s) of Algebra II; the OpenStax site lists sequences and series, other sites list complex numbers or trigonometry, and I personally recall learning about matrices and probability. Practical concerns such as the fact that students typically take geometry between these two courses and have significant cognitive growth over the multi-year period account for some of the topic overlap.

An overarching goal of both courses is to build students' capabilities to manipulate algebraic expressions, solve equations, graph functions, and translate word problems into one of the aforementioned three basic tasks. Why are the courses structured the way they are? To illustrate the relevance of and give students more practice carrying out the manipulating, solving, and graphing skills in different contexts. A major theme is that one can manipulate different algebraic expressions to highlight specific features (e.g., transform quadratics into vertex form to find an extremum, factor to identify roots) and use the equation-solving approach, graphing approach, or both to understand expression behavior and solve problems.

A drawback of this structure is that the topics can feel disconnected from one another, like lists of "prescribed rules" for manipulating different expressions and different procedures to follow in different contexts. My experience was that all these topics were unified in the subsequent Pre-calculus course which defined some elementary functions and the concept of inverses, developed coherent frameworks for graphing these functions in terms of scaling and translation and for solving equations in terms of some basic properties of equality, and focused the second half of the course on complex numbers and trigonometry. Emphasizing the commonalities among the topics and reinforcing the problem-solving approaches (manipulating, solving, and graphing) are some things you could do to help students at this level as they practice.

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As Steve says, there is no standard curriculum for Algebra 1 or Algebra 2 in the United States. However, if you intend to work with public school students in a particular U.S. state, you can refer to the state department of education's website. There, you may be able to find course descriptions, sample syllabi, instructional guidance, etc. that can help you understand the content and structure of the courses as intended by state-level decision makers.

For example, here are the chapters on Algebra 1 and Algebra 2 from the 2013 California Mathematics Framework. It's due to be revised soon, but there have been several delays.

As another example, here are Florida course standards for Algebra 1 and Algebra 2, current as of 2023. You can click on each standard for more detailed information and instructional guidance.

If you want a non-state-specific reference, College Board also sells "Pre-AP" Algebra 1 and Algebra 2 courses, and course guides are available online. These courses were developed relatively recently, but the College Board has strong influence on U.S. education, so I expect adoption to continue to grow.

Because of the lack of standardization, as well as a host of other reasons, it's hard to give a definitive answer to the question of why these courses are structured the way that they are. What I'd like to do instead is provide some examples of the variety of purposes and conceptions of school algebra that influence curricula.

Usiskin (1988) identifies four distinct conceptions of school algebra and relates them to different uses of variables. Here is a reproduction of a table from the article.

Conception of algebra Use of variables
Generalized arithmetic Pattern generalizers (translate, generalize)
Means to solve certain problems Unknowns, constants (solve, simplify)
Study of relationships Arguments, parameters (relate, graph)
Structure Arbitrary marks on paper (manipulate, justify)

From a historical perspective, Kanbir et al. (2017) identify six purposes of school algebra in their analysis. Four of the purposes line up with Usiskin's conceptions; there are two that are distinct:

  • Algebra as preparation for further mathematics and science; and
  • Algebra as a gatekeeper for entry to higher studies.

Debates over the relative importance of different purposes have gone on for more than a century and have influenced several waves of curriculum reform, so it's not uncommon for a course to feel incoherent or aimless. If you have specific students in mind, my recommendation is to go through their curriculum yourself, as well as any instructional resources that might be relevant, and to meet students where they are. That is, find ways to help students make sense of the material in alignment with purposes that are important for them.

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