You can find proofs by making analogies in almost any direction, but it helps to know which direction leads towards solid intuition the student already has and can build on, rather than just leading to more abstract nonsense that's equally unfamiliar.
In the case of having no intuition about algebraic manipulation, you can suggest a geometric interpretation to connect to intuition that's more likely to be there. For 2xy != 2x2y, draw two xy rectangles and one 2x by 2y rectangle.
Now the students all see the problem. Now they just have to connect the geometric intuition back to the algebra. This helps motivate the algebraic rules and shows why they must be what they are. Just the idea that geometric intuition exists -- that you can solve problems by putting pictures together in your head -- this isn't something every incoming freshman already consciously knows they have as a technique always available to them.
(This is just a wordy re-telling of Polya's "Draw a figure", from How to Solve It; if you haven't read, drop everything and get a copy.)
In the case of having no intuition about algebraic manipulation, you can suggest a geometric interpretation to connect to intuition that's more likely to be there. For 2xy != 2x2y, draw two xy rectangles and one 2x by 2y rectangle.
Now the students all see the problem. Now they just have to connect the geometric intuition back to the algebra. This helps motivate the algebraic rules and shows why they must be what they are. Just the idea that geometric intuition exists -- that you can solve problems by putting pictures together in your head -- this isn't something every incoming freshman already consciously knows they have as a technique always available to them.
(This is just a wordy re-telling of Polya's "Draw a figure", from How to Solve It; if you haven't read, drop everything and get a copy.)