Ordinary continued fractions give approximations by elements of Q to elements of R, the standard completion of Q. When K is a field, the field of rational functions K(x) has a valuation corresponding to vanishing at the point at infinity of the projective line over K. One easily constructs continued fractions giving approximation by rational functions to elements of the corresponding completion. Hyperelliptic curves have function fields which are degree two extensions of $K(x)$, so that these fields can be given by elements of the aforementioned completion. Using this, we have produced new examples of low genus curves whose Jacobians have elements of specified order. I’ll sketch some of the theory and practice of this elementary approach. This is joint work with K. Daowsud.