This article makes a case for introducing moving averages into introductory statistics courses and contemporary modeling/data-based courses in college algebra and precalculus. The authors examine a variety of aspects of moving averages and draw parallels between them and similar topics in calculus, differential equations, and linear algebra. The article: (1) shows how to calculate the moving averages for a set of data; (2) discusses the effect of the length of the moving average cycle on the sequence of values generated; (3) discusses the mathematical properties of the moving average as a mathematical function/operator; (4) addresses the problem of how to retrieve the underlying sequence of data values from the known sequence of values of the moving averages; and (5) discusses the significance of the moving averages in the sense of smoothing out the original data to better identify the pattern in that data.