Human essay grading is a laborious task that can consume much time and effort. Automated Essay Scoring (AES) has thus been proposed as a fast and effective solution to the problem of grading student writing at scale. However, because AES typically uses supervised machine learning, a human-graded essay corpus is still required to train the AES model. Unfortunately, such a graded corpus often does not exist, so creating a corpus for machine learning can also be a laborious task. This paper presents an investigation of replacing the use of human-labeled essay grades when training an AES system with two automatically available but weaker signals of essay quality: word count and topic distribution similarity. Experiments using two source-based essay scoring (evidence score) corpora show that while weak supervision does not yield a competitive result when training a neural source-based AES model, it can be used to successfully extract Topical Components (TCs) from a source text, which are required by a supervised feature-based AES model. In particular, results show that feature-based AES performance is comparable with either automatically or manually constructed TCs. [This paper was published in: "Proceedings of the 16th Workshop on Innovative Use of NLP for Building Educational Applications," Association for Computational Linguistics, 2021, pp. 85-96.]