Representations of numerical value have been assessed using bounded (e.g., 0-1000) and unbounded (e.g., 0-?) number-line tasks, with considerable debate regarding whether one or both tasks elicit unique cognitive strategies (e.g., addition or subtraction) and require unique cognitive models. To test this, we examined 86 5- to 9-year-olds' addition, subtraction, and estimation skill (bounded and unbounded). Against the measurement-skills hypothesis, estimates were even more logarithmic on unbounded than bounded number lines and were better described by conventional log-linear models than by alternative cognitive models. Moreover, logarithmic index values reliably predicted arithmetic scores, whereas model parameters of alternative models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a unified framework for numerical estimation with high descriptive adequacy and yields uniquely accurate predictions for children’s early math proficiency.