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Does anyone know of links to resources to explain why basic algebra rearrangement operations take place in a certain order?

A simple, seemingly absurd example, but not uncommon follows.

Say the student is tasked to make "y" the subject of:

x = 3y - 7

Now "obviously" the first step is to add 7 to both sides. But I see students whose first step is to try to divide by 3 - maybe because that's what they need to do to "get y on its own".

Or teaching a rearrangement of, say

3xy = 7x - 4y

to make "x" (or "y") the subject.

Weaker students find it difficult to know where to start and often try actions like "divide by y" rather than rearrange and apply linear factorisation (a very challenging step for weaker students).

Now, I'm not thinking the process of rearrangement can be reduced to a simple algorithm but are there any resources that can help the student to think "well, first I do this, then I do that, then I do that"? All the examples I can find example what to do in a particular situation but don't go into the why.

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    $\begingroup$ What's wrong with first dividing by $3$ if you want to solve $x = 3y-7$ for $y$? $\endgroup$ Apr 21, 2023 at 15:14
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    $\begingroup$ @JochenGlueck They forget to also divide $x$ and $-7$ by $3$. $\endgroup$
    – shoover
    Apr 21, 2023 at 20:24
  • $\begingroup$ well that's not related to the order, that's just forgetting to properly divide both sides by 3 $\endgroup$ Apr 25, 2023 at 23:39

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This is a golden opportunity to not provide such a resource.

Consider: have many pupils/students solve the equation, go around and see what they have done, and then have several of them present their own solutions side by side. Then have the students/pupils compare and contrast - which was easiest, shortest, most productive, where are there mistakes, and so on.

You can offer a strategy or two, too, if necessary, if only as foils, though you do risk students/pupils thinking that what you provided is the correct solution.

The idea here is that discussing different strategies shows the learners that there are many ways to approach a problem, and can get them to reflect what they have done. Also, since you are not offering a canonical choice, there is fairly low identity threat (thinking one is perceived to be stupid or do a wrong thing). Try to catch and find as many different strategies as possible, and tell this to the students/pupils, too.

This should help with your actual problem (the students/pupils have not figured out equations yet), and also with your perceived problem (increased strategy awareness should help with picking a good one, too).

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For a linear equation like $x = 3y - 7$, there is a simple algorithm, which in my classes I call the "general solving process". The general rule is, apply inverses in reverse order of operations. That is: when using inverse operations, everything works in reverse, including the ordering.

Usually I give an analogy of building a house (put up foundation, walls, roof) versus doing demolition of a house (take down roof, walls, foundation).

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    $\begingroup$ My analogy is socks and shoes. (Socks on first, then shoes, therefore shoes off first, then socks. We are undressing the variable, which I would not say with younger students.) $\endgroup$
    – Sue VanHattum
    Apr 23, 2023 at 22:19
  • $\begingroup$ @SueVanHattum: Nice, I hear that. Personally the analogy that clicked for me in 8th grade was two armies killing equal numbers on each side. The first time I uttered that as a TA teaching a college class, there were horrified looks everywhere, so from then on I kept it to myself. $\endgroup$ Apr 24, 2023 at 1:02

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