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I'm currently working on some precalculus packages for students who need review. For inspiration, I'm looking at some prealgebra books and I'm wondering why isolating for $x$ is taught before factoring.

I agree that there are ways to solve for $x$ when it comes to linear and quadratic, as seen below:

Solving for $x$ for linear: \begin{align*} 2x+4 & =10 \\ 2x & = 6 \\ x & = 3 \end{align*}

Solving for $x$ for quadratic:

\begin{align*} x^2+5x+6 & =0 \\ \left(x+\frac{5}{2}\right)^2 -\frac{25}{4} + \frac{24}{4} & = 0 \\ \left(x+\frac{5}{2}\right)^2 & = \frac{1}{4} \\ x + \frac{5}{2} = & \pm \frac{1}{2} \end{align*} $$x_1=-3, x_2=-2$$

But then, for cubic and quartic, it would be impractical.

On the other hand, if it was solving by factoring from the start, it seems easier to generalize to higher degree polynomials.

Factoring for linear: \begin{align*} 2x+4 & =10 \\ 2x-6 & = 0 \\ 2\left(x-3\right) & =0 \end{align*}

$$x=3$$

Factoring for quadratic:

\begin{align*} x^2+5x+6 & =0 \\ \left(x+3\right)\left(x+2\right) & = 0 \end{align*} $$x_1=-3, x_2=-2$$

Since the factoring version seems to be more flexible (since it could also apply to trigonometric functions), then why isn't that one taught first?

For my goals, since it's review, I'd probably introduce the two methods alongside each other but then state that factoring is probably the preferred method when it comes to higher courses.

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  • $\begingroup$ You cannot always use factoring when solving for x. Example when you want to solve $2^x=5$ for $x$. Solving for x is something most students will encounter more often and in more setting than factoring a polynomial. $\endgroup$ 17 hours ago
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Pedagogically speaking, factoring is a lot less intuitive than 'simple' rearrangement. For your example we have that, $$ 2x +4 =10. $$ When first teaching Algebra, there are many nice and neat tricks/visualizations to understand the process of unraveling the equation to solve for $x$. A classic analogy is to see the equation as a kind of seesaw that's balanced and you need to do steps so that the see-saw is always balanced.

A (perhaps insensitive) way that I was taught was to role play as a greedy family lawyer who had to 'divorce' $x$ from it's current relationship with the numbers it is with by doing actions that oppose what holds their relationship together in the first place.

So there are a lot of ways to explain this to someone who's first learning about something pretty abstract. In comparison, to get the solution by factoring does not have any nice analogy that can be used. In essence, we need to ask: what value of $x$ must be satisfied such that the RHS is zero? Which to an untrained mind is an extra layer of abstraction that doesn't need to be added until they're already comfortable with manipulating equations.

Further, it's even harder with the factoring method because the solution splits off into two 'branches'. Which is once again not immediately obvious to students why that should be the case. This problem can be swept under the rug with $\pm$.

However, I guess there are things that need to be untrained with the method of straight manipulation. So I agree that a greater class of problems can be solved a lot more straightforwardly by factoring, especially when working with $\mathbb{C}$.

Regardless, I think that usually people have a hard time learning Algebra for the first time and so we need to make this initial step up into abstraction as easy as possible.

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    $\begingroup$ I find the "greedy lawyer" story to be problematic in a lot of ways, and would not suggest using it with children (whose parents could very well be going through a divorce). $\endgroup$ Nov 4 '20 at 14:01
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    $\begingroup$ @StevenGubkin Yes that is very true. I didn't realize how insensitive that may be. Just mentioned it to convey the myriad of ways that isolating $x$ can be taught. In hindsight, I should have chosen a different example. $\endgroup$
    – Alias K
    Nov 5 '20 at 0:01
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This is kind of an interesting question. Three observations spring to mind.

First, you're really not going to short-circuit the need to present the basic inverses method of solving (the addition and multiplication) properties. Your factoring examples have skipped presenting those steps, but they're still in there, e.g.:

$$2(x-3) = 0 \implies x - 3 = 0 \implies x = 3$$

You definitely need to explain and justify that middle step (in which you'll add 3 to both sides of the equation). I mean: sometimes test-preparation materials can cheat on this, maybe just teach by rote that if you see $(x-3)$ as a factor you'll get a solution of $3$, but that would be invalid mathematics and students would suffer later on with that kind of "faith based math".

So you still need to deliver the basic inverses technique even to finish off your examples of solving by factoring. If you expect to also teach the method of factoring and the zero-product property, then at that point it seems like an unnecessary detour just to solve a linear equation.

Second, you can be sort of tricked by lots of "nice" examples that are being given to make life easy for the beginning students. Sure, many starting examples will have the constant term divisible by the linear coefficient (i.e.: factorable in integers, which is another unstated assumption). But what about any other case? E.g.: $2x - 3 = 0$? Again, you immediately need both the addition and multiplication principles in order to finish that off. What about general numerical problems: arbitrary fractions for coefficients, arbitrary decimals, etc.?

Note that many or most algebra books quickly exercise students on such general linear equations, ones that cannot be factored in integers. At this point you have a fairly nice general technique for solving linear equations of all sorts. For example, see OpenStax Elementary Algebra, Section 2.5: "Solve Equations with Fractions or Decimals", which comes immediately after the general strategy for solving linear equations by inverses.

Third, many books and curricula also treat general linear inequalities at about the same time. That's pretty close to the same process, with one added trick (flip inequality direction if one multiplies by a negative number). It's even less clear what kind of trick you'd apply to jump over that "missing" step in your examples to handle this with an always-factoring approach. Again, see OpenStax Elementary Algebra, Section 2.7, for these applications.

(Note also that this curriculum then follows with graphing lines and solving linear equations before higher-degree objects are handled; this provides a spiral-type path where you get to revisit the ideas of solving equations, inequalities, and graphing, in progressively more advanced contexts -- which is often needed by such basic students.)

In short, the general process for solving linear equations and inequalities can pretty quickly be presented, and in fact must be presented, even if you wanted to focus on factoring all the time (which therefore presents an unnecessary delay). So the student has a fairly nice package of tools to handle linear stuff, possibly numerically with calculator technology, even if it's not factorable in integers.

In fact, for some students they may not progress any further at all in their math pathway. Consider in this case OpenStax Prealgebra: that work manages to cover solving linear equations, but never gets to any higher-degree work. For some students that will be the end of the line, and time spent on factoring will be an unhelpful delay and distraction. (A key point of debate for basic math skills at my institution has in fact been administrators arguing that non-STEM students don't need to learn factoring, for example.)

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    $\begingroup$ I agree with much of what you are saying here. I just wanted to point out that solving equations like $x-3 = 0$ "by inspection" rather than by adding 3 to both sides is valid, and I would claim important for understanding. I would also like to see students solving things like $x+49 = 48+7$ by inspection without needing to first add $48+7$. Students might solve this mechanically without ever realizing what they are trying to do: find a value of $x$ which makes the equation true. Doing some problems in "ad hoc" ways forces you to grapple directly with the meaning of solving an equation. $\endgroup$ Nov 5 '20 at 13:05
  • $\begingroup$ The phrase "by inspection" has always struck me a bit strange in the context of solving an equation, because it seems to just indicate “mentally, without writing anything”. In the case of identifying that $x=4$ is a solution to $x(x-4)=0$, I often get students who appear to be doing this kind of mental solving by inspection, but they are really just writing down the opposite of a particular character in the equation — something I observe when they write down “5” as the solution to $x(3x-5)=0$, or, remarkably, “x” as the solution to $x(6-x)=0$. $\endgroup$
    – Nick C
    4 hours ago
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With others, I think the mechanical aspects of adding etc. to sides of the equation is easier at first. And super helpful in really driving the concept of the equality idea. When you do it in detail, you have a set of steps where you add, divide, etc. on both sides of the equation. NOT "flipping over the equals and reversing sign"--that is sort of a quick way to think of it later.

So 2x + 5 = 6 - 7x

[Add 7x to both sides.]

2x + 5 +7x = 6 - 7x +7x

[Combine terms in x.]

9x + 5 = 6

[subtract 5 from each side.]

9x + 5 -6 = 6 - 6

[combine the numbers]

9x - 1 = 0

[add one to each side]

9x -1 + 1 = 0 + 1

[combine the numbers]

9x = 1

[Divide both sides by 9.]

9x/9 = 1/9

[Do the division.]

x = 1/9

Kids should actually write this whole thing down (not the bracketed comments, but the step by step solution). It's very powerful for them to learn this process. Remember that "x" is a scary thing. We're doing math with letters in it now!!! So doing this repeated, clear manipulation is useful for them to start understanding equations and how to manipulate them.

Factors are kind of a short cut. And not always easy to identify by eye. So teach them the methodical method first. Then when having a general comfort with letters in math and with manipulations, they can move to factoring.

Plus if you have exes and numbers on both sides of the equals, you're going to need to do some manipulation anyway, even if you factor later. But I would still teach the brute force step by step method first.

P.s. The "by inspection" stuff is fine and I do it and the sharper kids jump to it also. (What Feynman called solving it with arithmetic instead of algebra.) But it's not always easy to do this with a complicated expression. And there's nothing to stop going over this stuff later. But one thing at a time. Especially with weaker students.

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