Earlier this week, a teacher posted in the College Board’s AP Calculus Community a request for an explanation of computing the arclength of a curve without relying on formulas.
The following video is my proposed answer to that question. In it, I derive the fundamental arclength relationship before computing the length of from x=0 to x=3 four different ways:
- As a function of x,
- As a function of y,
- Parametrically, and
- As a polar function.
In summary, the length of any differentiable curve can be thought of as
where a and b are the bounds of the curve, the square root is just the local linearity application of the Pythagorean Theorem, and the integral sums the infinitesimal roots over the length of the curve.
To determine the length of any differentiable curve, factor out the form of the differential that matches the independent variable of the curve’s definition.
I’ve been in several conversations over these first couple weeks of school with colleagues in our lower and middle schools about what students need to do to convince others they understand an idea.
On our first pre-assessments, some teachers noted that many students showed good computation skills, but struggled when they had to explain relationships. Frankly, I’m never surprised by revelations that students find explanations more difficult than formulas and computations. That’s tough for learners of all ages. But, in my opinion, it’s also the most important part about developing a way to communicate mathematically.
In the other direction, I frequently hear students complain that they just don’t know what to write and that teachers seem to arbitrarily ask for “more explanation”, but they just can’t figure out what that means.
Just like writing in humanities classes, a math learner needs to seriously consider his “audience”. Who’s going to read your solution? I think too many write for a classroom teacher, expecting him or her to fill in any potential logical gaps.
Instead, I tell my students that I expect all of their explanations to be understandable by every classmate. In short,
Don’t write your answer to me; write it to a friend who’s been absent for a couple days.
If a random classmate who’s been out a couple days can get it just based on your written work, they you’re good.