That's the goal is math and science, too. Too often kids get stuck on and obsessed with the tools (prefixes, scientific notation, algebra, trig, calculus, definitions, moles) and miss the forest for the trees. Having to spend a lot of mental energy on those things also means that you simply don't have the stamina to get to the end of something with a lot of sub-parts, because the sub-parts require too much effort (because you can't do them fluently!).
We worked at this fiendish challenge problem from The Physics Teacher today.
Mainly, I wanted to illustrate the concept of looking at different cases or regions within the problem. Here, we had to consider cases in which neither block slipped, only the bigger block slipped, or only the smaller block slipped. Cases in which both blocks slip are ruled out by the ribbon's masslessness (think about it - it's a little subtle!).
We went through what the accelerations of each slipping block and non-slipping block (+ribbon) would be in each of the three cases, as well as what the static friction force on each non-slipping block would be in each case. If a scenario is impossible, you'll get an impossible result from the comparison of the net force required for the acceleration indicated and the acceleration that the static friction /gravity can provide.
It looked something like this:
That's quite a bit of algebra, but... how long did it take us to come up with these functions for all of these different cases? 10 minutes, maybe!
The reason is fluency. These kids would've taken forever to get these out last year, and would've made a lot of wrong turns and not noticed for a while, because they were still stuck with partial fluency on some of the skills needed to write these equations. By my estimation, these are the skills that the kids had to be fluent with to model these three situations so quickly:
- Identification of forces (normal, mg, kinetic and static friction)
- Identification of acceleration direction and which accelerations will be equal to which others
- Trig and vector components
- Rotation of axes
- Defining positive in a consistent way for both blocks
- Newton's 2nd law
- Requirements for static friction to be in effect
- Symbolic algebra
- Relationships between the friction coefficients
After all, no composer can write an opera if they have to do "Every Good Boy Does Fine" to know what the notes are!