Dependent t-test or something else when pre- and post-scores use different scales? What would be the most appropriate statistic to use if one wanted to see whether an instructional treatment produced learning gains in a classroom in which all students were tested before and after the treatment? Assume the learning measures are valid, but the pre-test has, say, a score that can range from 0-10 but the post-test can go from 0-100 (imagine a pre-test that's a short quiz and a post-test that's a longer test, for example).
I recognize the subjects represent matched pairs. If various assumptions are met, it seems like the null hypothesis could be tested with a matched-pairs t-test? Is that right? If so, what scores (numbers) are used in the actual test, given how different the two scales are? Do you have to do something like transform each raw score into a percentage?
 A: You cannot determine whether a treatment has an effect without a comparison to a control group and you cannot use the pre-treatment scores as the control, at least not convincingly.
Let's elaborate on your hypothetical situation: You give students a test at the start of the semester, give them the treatment, then test them again at the end of the semester. You cannot use the pre-treatment score as the control for the post-treatment score because students will learn something during the semester with or without the treatment. In other words, the effect of the treatment is confounded with the fact that students learn when they go to school.
Since this is a hypothetical experiment, however, suppose that you have pre and post measurements for a control group as well (eg. another class in the same school).
You can model the data you've collected with a regression: the predictors are the pre-test (X), an indicator variable for whether a student received the treatment or not (T) and any covariates (Z); the outcome is the post-test (Y). The effect of the treatment is the coefficient $\beta_T$ of the treatment indicator and you can test whether $\beta_T$ is greater than zero.
Regression it doesn't assume that the slope of Y on X is 1; instead it estimates it. This means that regression elegantly handles the case when pre and post scores are on a different scale, unlike gain scores (gain = post score - pre score) which implicitly assume that the slope is 1.
You can learn more about how to estimate a treatment effect in Chapter 19 of Regression and Other Stories [1]. The textbook uses an educational study as a running example and discusses many subtleties of performing and analyzing such an experiment (eg. one treatment and one control classroom won't be enough because the treatment effect might be confounded with a teacher effect.)
[1] A. Gelman, J. Hill, and A. Vehtari. Regression and Other Stories. Cambridge University Press, 2020.
