As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years. Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more failure.
The root of the problem, in my experience, is the way algebra is taught as something alien, and in particular, different from arithmetic, which it really isn’t (at least in the early years).
So first off, constant emphasis on the fact that “$x$ is just a number you don’t know yet”. So it behaves like a number, and you can do all the stuff to it, that you can do to numbers.
Next, the nature of equality $2 + 3 = 4 + 1$.
And from there, the fact that when you do the same thing to both sides, you still end up with two things that are equal.
Always “do the same thing to both sides” (since this is clearly based on the nature of equality), never “move this from one side to the other and change the sign” (which is a magic rule that makes no sense until you have a deeper understanding).
Once you get them happy with the idea that doing the same thing to both sides is the way to go, you can give them suggestions for which things to do in which order, but stress that provided they rigorously write down the consequence of the thing they decide to do to both sides, they won't go wrong (although some ways are harder – look out for these as a pointer that choosing another way will be easier).
The manipulation of each line is easy, once you’ve got them to decide what they’re going to do at each stage.
For instance, in the example, $3x + 5 = 14$:
- First, decide what to do to both sides (subtract 5)
- Write down first what you have ($3x + 5$), then do what you’ve decided. So you get $3x + 5 - 5$, and on the RHS, $14 - 5$.
- Then collect terms and simplify to get $3x = 9$.
- Then repeat for division by 3.
Emphasise that once you’ve decided what to do at each stage, there’s very little thinking, since you’re just writing – starting with what you had on the previous line, and adding on the chosen operation.
Figuring out what to do (add or multiply, subtract or divide) needs to come after they are truly grounded in the principle that doing the same thing to both sides is the key.
They will also need help with things like why $3x/3 = x$. Again, use numbers to illustrate, and stress that $x$ is just a number, so it behaves the same way as a number.