Differential Geometry general question

Ok, in general, I know that if the coefficients of the First Fundamental Form agree for two surfaces parameterized by X and Y, the the map X(Y^-1) is an isometry, or the two surfaces are isometric.

I also know that if two parameterizations don't have the same coefficients, this does not imply that the two surfaces are not isometric.

So i have two parameterizationsthat have different coefficients of the FFF (first fundamental form) but have the same Gaussian curvature. I need to prove that the two surfaces are not isometric. (ie i need to prove that the converse of Gauss's Great Theorem is false).