This is an example of only one kind of "linearity." I don't think it's been mentioned yet.

Whenever I write something like $$\dfrac{2x+3}{2}$$ on the board, somebody will inevitably say, "Cancel the twos!!" And I respond, "Wait! So that means five divided by two is three, right?" $$ \dfrac{5}{2} = 3 $$ "because" $$\dfrac{5}{3}=\dfrac{2+3}{2}=3$$ "Right??"

And then I proceed with showing them the way to factor and cancel in an expression such as

$$\dfrac{4x+6}{2} = \dfrac{2(2x+3)}{2} = 2x+3$$