What are the pros and cons of teaching the distance formula? I can think of two of each:
- Pro #1: It’s useful and convenient.
- Pro #2: Future teachers may expect your students to know it.
- Con #1: It’s nothing but the Pythagorean Theorem and is therefore unnecessary.
- Con #2: Experience shows that many students memorize it and then recall it incorrectly.
One of the Big Ideas we try to teach in mathematics is the idea of abstraction. (That’s why we math teachers are not especially fond of units, by the way.) But it’s not clear whether teaching the distance formula enhances or contradicts the development of abstraction. On the one hand, it encapsulates a lot of different (concrete) instances of finding the distance between two points in the plane. On the other hand, it loses the abstraction that we’re really just talking about the Pythagorean Theorem.
Tony Asdourian has an excellent post on the subject. Here is an excerpt, but do check out the entire post:
I think we should ban(!) teaching the “Distance Formula”, at least for the large majority of 9th and 10th graders. Why, you might ask, if I am trying to teach them the value of abstraction? Because time and time again, students who come in to my class “knowing” this formula have no idea where it comes from. And because for the ways in which it is used in the first two years of high school, the Pythagorean Theorem is much more intuitive and direct. Kids never have a problem remembering from middle school that a2 + b2 = c2 for right triangles, and if you ask them to find the distance between two points by drawing an appropriate right triangle and making calculations, they can do it easily.