Dealing with g in conceptual physics - and a "false calculation" example

A correspondent has questions about how I deal with g in conceptual physics:

I notice that you intentionally use g as 10 N/kg instead of m/s/s and say that weight is the force of a planet.  I've noticed that when we get to energy, students struggle with g for GPE=mgh. They all want to put the mass (say 0.5 kg) but then they say 5N or 5N/kg for g.  It seems they have a very shallow understanding of what we're doing with 10 N/kg (and I notice you don't tell them explicitly to do mg for weight).  And the other thing they do is anytime they see a mass in kg, they'll "convert it to newtons." Or they'll see a force of 5 N and say it equals 0.5 kg.

Do you care whether they make the connection between "objects in free fall gain/lose 10m/s/s" and the gravitational field? For instance: when they're solving for speed based on KE, when N/kg doesn't fit with the other units. Or do you not really care about manipulating units, as long as they know what units go to what quantities (i.e. speed is always m/s, energy is always J).  

In conceptual, I don't care at all if they use careful language about g.  I never explicitly make the connection between the 10 m/s/s free fall acceleration and the 10 N/kg gravitational field.  They do need to put correct units on, of course... but I am fine with them "converting" 5 kg to 50 N.  As long as they sort of can get the correct values for weight and mass, I'm fine.  No unit manipulation ever.  

If they can do a calculation with mgh in a table, indicating correct units on m, g, and h, with J on the final answer, this is fantastic (and difficult for conceptual students).  At this point if they plug in numbers correctly in a table and get a unit wrong here and there, as long as the final units are right, I'm fine.  

Usually, I'm asking for a comparison between two situations, like "what happens to the potential energy of this 0.5 kg object when its height doubles?"  Here many students like to fall back on what they call a "false calculation" - plug in simple values to the formula for the first situation, then the second situation, and see how the results differ.  See below.  

Things to notice: 

* This student made the mass 1 kg to make the calculations easier.  I encourage that!  This is the conceptual physics version of "Bertha's Rule of Ones."  They chose 1 m and 2 m to double the height. If they had chosen 3 and 6 m, that's fine, too.

* I do expect units on their tables; I don't expect units on their arithmetic, because a lot of 14 year olds I teach have trouble multiplying (1 kg)*(10 N/kg), even though they can multiply (1)*(10) just fine.  Make the process simple, and likely to lead to the right answer.  

* Yeah, this student forgot units on g in the second table.  But they got 'em in the first.  I would likely let this go.  I mean, there's a line in the sand between doing math with no physics on one side, and pedantry on the other.  The goal is that by year's end, my students themselves should be uncomfortable if they left units off something.

What about in AP Physics?  I still start out not caring much about the true meaning of g... still no unit manipulation.  They do eventually need to understand that free fall g is the same as gravitational field g, because the AP exam discusses the difference between gravitational and inertial mass.  Yet, I don't even mention that difference or the connection until after spring break.  I'm far more concerned with their articulation of correct annotated energy bar charts; and with the conceptual difference between mass and weight.  (In AP, I do encourage students to write mg for the force of the earth on an object.)

You're right that student understanding of 10 N/kg is often quite shallow.  Usually they get it more as the year goes on; sometimes they don't truly "get it."  But if they can make proper energy bar carts and make correct qualitative and quantitative predictions with energy bar charts, then understanding the meaning of g can be left for a future class - or might never happen.