Variable Parts: A New Perspective on Proportional Relationships and Linear Functions

We present a mathematical analysis that distinguishes two quantitative perspectives on ratios and proportional relationships: Multiple Batches and Variable Parts. We argue that (a) existing research on proportional relationships has addressed Multiple Batches but has largely overlooked Variable Parts, (b) Multiple Batches makes the co-variation aspect of proportional relationships more explicit, while Variable Parts makes the fixed multiplicative relationship between two quantities more explicit, (c) the distinction between Multiple Batches and Variable Parts is orthogonal to the within-measure-space versus between-measure-space ratio distinction, and (d) Variable Parts affords promising new approaches for addressing linear relationships. [For the complete proceedings, see ED597799.]