I'm supposed to come up with two ways to introduce Newtons method for the approximation of zeros for highschool students. (That is the method using tangents and with the formula $ x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})} $). My first suggestion is to describe the method geometrically. Then the students should first understand this geometric description and then deduce the formulas on their own.
Now I'm struggling to come up with a second approach. My only ideas were (1) to describe the secant method and/or the regula falsi method (geometrically) and then somehow push the into the right direction so that they come up with Newton's method. But this would cost too much time. Or (2) to describe the method algebraically and they should understand and describe what happens geometrically. But this approach seems somehow forced and arificially to me.
So my question is, has anyone an suggestion, how to introduce Newton's method in a non-geometric way? I'm thankful for any idea.
You might look at the special case of solving $x^2 = a$, where $x_{n+1}$ is the average of $x_n$ and $a/x_n$. Then you can see that the "error" $a - x_{n+1}^2$ is a constant times the square of the previous error $a - x_n^2$.