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I know pre-algebra is like a terminology thrown around to really basic stuff that are taught before high school algebra. Some stuff taught there are already considered as part of algebra in some sources. I just want to ask what are the topics considered to be part of this, whether they are common or come only in some situations.

Yes, I know this will sound again as some opinion based thing and someone will point out that I should consider it on my own. I am asking here for what are the general stuff. Yes, even if it is based on your opinion but somewhat backed up by sources.

I wanted to ask this to get a grasp on what topics should people know before taking algebra in high school and which part of those don't intersect much with high school algebra. Like for example, arithmetic are stuff that are emphasized in elementary. Sure, variables are used in algebra but the introduction of what the operations are were totally done in elementary.

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I think the most important concepts to have mastered before starting algebra are basic arithmetic and general reasoning. Fluency with negative numbers and the order of operations cannot be under-estimated as this prevents troubles with substitution. Students should understand what is meant by equality and inverse operations, i.e. that $70 + 2 = 72$ is equivalent to $72 - 2 = 70$.

In the UK, students start algebra in primary school but tend to get their first proper exposure when they're around 11 or 12. At this stage, many still need more work on their numeracy. The difference is clear: students who have stronger numerical reasoning tend to be better at generalising to algebraic methods. Things like working out how much change does one get given a shopping list and prices or problems that work backwards to find 'what number did I start with' are examples of things young students would benefit from being confident with.

I've seen teachers trying to teach algebraic expressions to students with poor numerical reasoning. For example, if there are 7 socks in a packet and I buy $x$ packets, then it is obvious for many that there are $7x$ socks in total, but for weak students that struggle with multiplication, the generalisation is harder.

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    $\begingroup$ Aren't pre-algebra and pre-calc fake subjects, invented by educrats to cover their butts for not teaching "basic arithmetic and general reasoning" well enough in elementary and middle school? (Sometimes pre-calc means trigonometry, but why would not they call it trigonometry then?) $\endgroup$ Commented Aug 15, 2020 at 17:41
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In my experience, students need to have a firm grasp of fractions and order of operations to succeed in algebra. Manipulating fractions with algebraic expressions can be difficult if you don't know how to manipulate them with numbers. Without a solid grounding in fractions, you can't simplify an algebraic fraction, add, subtract and multiply them, or find the domain. Order of operations is necessary for evaluating expressions, using a calculator (where do those parentheses go?), and figuring out how to solve an equation by working backwards. I'm sure there are other important topics but these two seem essential and many students are weak.

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As how precalculus is mainly the algebraic manipulation necessary to do calculus questions, I'd suppose that prealgebra is the arithmetic manipulation necessary to do algebra questions. Looking at the wikipedia page, it seems that it's mostly ability to work with numbers and variables with basic operations of $+$, $-$, $\times$, $\div$, (as well as basic roots and exponents). And this makes sense since how would a student balance equations well if they can't do the computations with the coefficients and constants.

I think one thing that's really lacking for students are inequalities. They always seem to know how to solve for something with direct relationships, but unsure about indirect relationships. For example, triangle inequality is used a lot in mathematical proofs and probably comes up quite a bit in high school level math contests, but I don't recall seeing it very often in actual lessons or textbooks.

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On a simple level, I remember,

  • Pre-algebra: linear equations in one variable, 7x +10 = 5-3x
  • first year HS algebra: lots of lines, y=mx+b
  • second year HS algebra: conic sections, exponents, logs

The manipulation work in pre-algebra to sort out and solve for the x, is a basic skill that is called on in many higher classes. Often re-arranging the spinach, to see clearly enough to apply some rule, is half the battle.

We had a heavy dose of inequality battling in a course called "functions", which along with analytic geometry came after Algebra 2, and served as pre-calc. I disagree with Robbie's comment about inequalities in the context of pre-algebra. That topic should come later.

[EDIT] Note that some elements of algebra are included in pre-algebra. Just as some derivatives and anti-derivatives are included in pre-calculus ("functions" and analytic geometry). Well...at least in Fairfax County public schools of the 70s and 80s. And Algebra 2 covered all the logs and such that are part of "college algebra". And there was a full semester on trig also (after algebra 2), but before "functions". The GT kids took Algebra 2/Trig (which was 3 semesters in 2).

But I would not rely too much on names. Looking at textbooks, I see several pre-algebra texts that have the "solve for x" as key content. It's like baby steps into algebra. Then they throw a y in along with it in algebra 1.

For example this is the first hit on google (that I got for a prealgebra textbook pdf).

https://www.cbsd.org/cms/lib010/PA01916442/Centricity/Domain/2723/PreAlgebra%20textbook.pdf

Note that they get right into solving for x in chapter 2 of the book. (See page 91 for the classic adding things to both sides of the equation stuff.) Granted, there's a bunch of pictures of girls with horses...don't rememember that when I did it. But "solve for x". Classic pre-algebra. Kid has exercised that before high school algebra 1.

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    $\begingroup$ Linear equations in one variable. This is a hallmark of beginning algebra. $\endgroup$ Commented Aug 14, 2020 at 14:18
  • $\begingroup$ For schools in my state (and era) and even surrounding states, 2nd year algebra focused a lot on factoring (mostly difference of squares and trinomial type; also covered in 1st year algebra, but the problems were simpler), rewriting rational expressions, quadratic equations (often reached at the end of algebra one, but not dealt with in much depth), two linear equations in two unknowns, and more difficult "applied problems" (mixture, rate, time/distance, etc.). As for conic sections, exponents, logs, trigonometry, graph shift methods, etc., this was done in "analysis/precalculus/4th-year-math. $\endgroup$ Commented Aug 15, 2020 at 18:52

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