Several researchers have dealt with the one-dimensional fractional heat conduction equation in the last decades, but as far as we know, no one has investigated such a problem from the perspective of developing suitable fractional-order methods. This has actually motivated us to address this problem by the way of establishing a proper fractional approach that involves employing a combination of a novel fractional difference formula to approximate the Caputo differentiator of order $\alpha$ coupled with the modified three-point fractional formula to approximate the Caputo differentiator of order $2\alpha$, where $0<\alpha \le 1$. As a result, the fractional heat conduction equation is then reexpressed numerically using the aforementioned formulas, and by dividing the considered mesh into multiple nodes, a system is generated and algebraically solved with the aid of MATLAB. This would allow us to obtain the desired approximate solution for the problem at hand.