This paper investigates the formation scaling control problem for multi-agent systems by utilizing the stress matrix associated with a universally rigid framework. Compared with the existing results on formation scaling control, we consider a more challenging scenario where only one agent has the knowledge of the desired formation size. To cope with this constraint, we first propose a distributed estimator for the remaining agents to estimate the scaling parameter. Then by employing the outputs of the estimator, we design a new class of formation scaling control algorithms for universally rigid frameworks such that the overall formation converges to the prescribed shape with the desired scaling. Numerical simulations are carried out to validate the theoretical results.