The "factor of change" method for semiquantitative reasoning - AKA Bertha's Rule of Ones

Bertha (pictured) is a
very smart elephant 

[Note: Bertha, who is a very smart elephant, uses this "factor of change" approach all the time.  Since about 2020, I've named this method after her: Bertha's Rule of Ones."]

"Semiquantitative reasoning" means problems asking things like "We double the amplitude of an object oscillating on a spring.  What happens to the object's maximum speed?" 

On one hand, you could make up values and solve the energy conservation problem for speed.  Do this twice, once with the amplitude doubled, and see how the result changed.  This is a straightforward method to teach.  The downsides are, it takes a long time for students to execute; it's all to easy for a student to make a calculational error; and most importantly, it's difficult for the student to extract what feature of the mathematics leads to the result, as the AP exam's "qualitative-quantitative translation" questions require.

So, solve in variables, you might say.  Good.  Yet students flounder with simple algebra, working themselves into a circular-reasoning state. "Converting potential to kinetic energy gives the equation 1/2kA^2 = 1/2mv^2.  Since amplitude is inversely proportional to spring constant, the spring constant goes down when amplitude increases.  Then mass is directly proportional to spring constant, so also decreases.  Now mass is inversely..."  Aargh! 

The more astute students solve for speed, getting A times root (k/m).  Great!  But then they start from the original equation again, plugging in (2A) instead of A.  The math that I've seen a thousand times:

Or,

Grrr.  This is what causes the good folks at my workshops to complain a la Gus the Theatre Cat.  ("And I say now these kittens, they do not get trained / in the algebra skills that we once all attained / They never do listen, they're dreaming of sex / And they think they are challenged by solving for x. Hah!)

I've avoided must frustration on my and my students' part by teaching the factor of change method.  This was taught to me about half a decade ago by someone in one of my summer institutes - I forget who, or I would credit her or him here.  (If it was you, please let me know and I'll update the post!)

In the factor of change method, you start by solving for the requested quantity just once, and in variables only:

Now, plug in a (1) for every quantity that doesn't change; and plug in a (2) for a quantity which doubles.  Or a (3) for a quantity which triples, or a (1/2) for a quantity that's halved, etc:

Look how the algebra difficulties have been cut through like a knife through hot tofu. The max speed is multiplied by 2.  Done.  

I do need to show students that a numerical value in an equation - like a 2 or a pi - cannot change, and so we plug in (1) in the factor of change method.  They get that very quickly.

What feature of the mathematics leads to this physical result?  Easy enough - the speed is related to the amplitude.  With k and m constant, doubling amplitude leads to a twofold increase in speed.  The factor of change method strips the mathematics down to the simplest, most comprehensible skeleton I can imagine.

Note: I updated this post when Brian Frank pointed out a math error in the original.  Grrr.  Thanks, Brian.  :-)