> An effective math teacher will point out that “rise over run” isn’t the definition of slope at all but merely a way to calculate it.
Depends on the level. For introductory algebra, or classes before calculus, where the "slope" concept is used exclusively with linear functions, "rise over run" is a perfectly adequate definition.
For calculus, you can rigorously define slope as the limit of (f(x+h) - f(x)) / h as h -> 0 (or any of several other equivalent definitions). The subtleties of the proper definition are likely to be more confusing than enlightening to a beginner.
But even the rigorous definition is still a "rise" term f(x+h) - f(x) divided by a "run" term h. The phrase "rise over run" is an easy-to-remember mnemonic which actually does a really good job of capturing the underlying idea without going deep into more technical issues involving limits (calculus) or the guts of exactly what a real number is (real analysis).
Once the linear case is completely understood and the student has some basic facility with both mathematical reasoning and algebraic manipulation, THEN is the time to introduce the more general definition of "slope". Even then, mathematics teachers tend to use the word "slope" mainly in the linear case, and "derivative" in the general case.
As far as I know, this has been the standard way to teach high school / early college level mathematics for decades.
> An effective math teacher will point out that “rise over run” isn’t the definition of slope at all but merely a way to calculate it.
Depends on the level. For introductory algebra, or classes before calculus, where the "slope" concept is used exclusively with linear functions, "rise over run" is a perfectly adequate definition.
For calculus, you can rigorously define slope as the limit of (f(x+h) - f(x)) / h as h -> 0 (or any of several other equivalent definitions). The subtleties of the proper definition are likely to be more confusing than enlightening to a beginner.
But even the rigorous definition is still a "rise" term f(x+h) - f(x) divided by a "run" term h. The phrase "rise over run" is an easy-to-remember mnemonic which actually does a really good job of capturing the underlying idea without going deep into more technical issues involving limits (calculus) or the guts of exactly what a real number is (real analysis).
Once the linear case is completely understood and the student has some basic facility with both mathematical reasoning and algebraic manipulation, THEN is the time to introduce the more general definition of "slope". Even then, mathematics teachers tend to use the word "slope" mainly in the linear case, and "derivative" in the general case.
As far as I know, this has been the standard way to teach high school / early college level mathematics for decades.