Its that new math. The supreme court redefined sqrt(4) as 1.68 just recently. So your code is correct. Now all we need to do is convince your teacher of that. :-)
The divide and average method for sqrt is pretty simple really. In fact, it converges pretty rapidly.
while abs(asqrt*asqrt - a) > tol
asqrt = (a/asqrt + asqrt)/2
So the idea is you divide the current estimate of the sqrt into a, and then average THAT result with the current estimate. Repeat until you get bored, or until it meet your tolerance. 1 is a good starting point.
What you will see here is this is actlually a quadratically convergent estimator of the sqrt. It does indeed converge rapidly, effectively doubling the number of digits in the estimate after each iteration. This is why I called the convergence behavior quadratic.