I can think offhand of about six situations in algebra-based introductory physics where negative signs are essential:
(1) algebraic kinematics, where displacement, velocity, and acceleration must include a direction
(2) momentum conservation, in which velocity must include a direction
(3) work done by a force
(4) all variables in the first law of thermodynamics (heat added to a gas, work done on a gas, change in internal energy)
(5) electric potential, where a negative potential means "below zero", and where a negative charge produces a negative potential
(6) thin lenses and mirrors, where negative image distance indicates a virtual image; and negative focal length indicates a diverging lens or mirror
For (1) and (2), I teach students to make a chart with all of the possibly relevant variables. On top of that chart they are supposed to indicate which direction they've chosen as positive. The chart helps de-emphasize the algebra.
If I ask for substitution directly into equations, students get overwhelmed. They try to substitute and solve all at the same time, losing track of what is known and unknown; a negative sign easily gets overlooked. Furthermore, the meaning of the negative sign gets completely lost. "Oh, the acceleration is negative. Okay, now the math works," is what I might hear. It's not clear why the acceleration should be negative, just that it gives the right answer. And conceptually, "negative acceleration" takes on an incorrect connotation, as if the acceleration were somehow bad, undesirable, or naughty.
The process of making and filling out a chart clearly differentiates problem solving into a physics step, and a math step. When a kinematics chart includes three of the five variables, the physics is done; everything else is mathematics. In practicing problem solving, students will certainly forget a negative sign occasionally. But because that math step is so obviously separate from the physics, I can train them to "debug" by checking signs with physical reality -- e.g, "the object was slowing down, so acceleration and velocity are in opposite directions. Whoops, acceleration must be negative here, then." The algebra is usually redone from scratch, yielding a correct answer with physical understanding included.
For (6), I teach the assignation of signs before I even introduce the thin lens equation. We practice ray diagrams using templates until everyone is really good at drawing. But a key part of each diagram is to measure the image distance, object distance, and focal length, recording each value in a chart. I'm clear that all entries in the chart require a sign -- if I see a value without a + or - sign, the value is not assumed to be positive -- it's assumed to be wrong.
Since I teach optics in March, my class has no trouble actually plugging into the relevant equation, and they're used to the chart approach. All I have to do is prod them to fill in the chart properly.
[In fact, I think that next time I teach the junior-senior course I will use this "required sign" approach even in kinematics and momentum: if I don't see a + or - sign, it's wrong.]
Numbers (3), (4), and (5) are much more difficult. Electric potential, work, heat, and internal energy are scalars, so the negative sign does not indicate a direction. I don't generally discuss the "meaning" of the negative sign in these cases; rather, I repeatedly emphasize how to figure out the sign:
* Work done by a force is positive when the force and the displacement are parallel. Work done by a force is negative when the force and displacement are antiparallel.
* Electric potential due to a positive charge is positive, and due to a negative charge is negative. Positive charges are forced high-to-low potential, and negative charges are forces low-to-high potential. A negative potential is just potential "below zero."
* In the first law of thermodynamics, ΔU is positive when temperature increases, negative when temperature decreases; Q is positive when heat is added, negative when heat is removed; and W is positive when volume decreases, negative when volume increases.
You might use negative signs in a bunch of other situations, too. I know. I don't. In a forthcoming post, I will explain how I avoid dealing with signs elsewhere, because I simply don't trust first year physics students with a negative sign.