This article explores teaching practices described in NCTM's *Principles to Actions: Ensuring Mathematical Success for All*. Common factors, common multiples, strip diagrams, and double number lines are discussed in this, the third installment in the series.

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- Author or Editor: Sybilla Beckmann x

### Andrew Izsák, Sybilla Beckmann and Torrey Kulow

### Sybilla Beckmann and Andrew Izsák

In this article, we present a mathematical analysis that distinguishes two distinct quantitative perspectives on ratios and proportional relationships: variable number of fixed quantities and fixed numbers of variable parts. This parallels the distinction between measurement and partitive meanings for division and between two meanings for multiplication—one rooted in counting equal-sized groups, the other in scaling the size of the groups. We argue that (a) the distinction in perspectives is independent from other distinctions in the literature on proportional relationships, including the within measure space versus between measure space ratio distinction; (b) the psychological roots for multiplication suggest the accessibility of the two perspectives to learners; and (c) the fixed numbers of variable parts perspective, though largely overlooked in past research, may provide an important foundation for central topics that build on proportional relationships. We also suggest directions for future empirical research.

### Andrew Izsák, Torrey Kulow, Sybilla Beckmann, Dean Stevenson and Ibrahim Burak Ölmez

We report results from a mathematics content course intended to help future teachers form a coherent perspective on topics related to multiplication, including whole-number multiplication and division, fraction arithmetic, proportional relationships, and linear functions. We used one meaning of multiplication, based in measurement and expressed as an equation, to support future teachers' understanding of these topics. We also used 2 types of length-based math drawings–double number lines and strip diagrams–as media with which to represent relationships among quantities and solve problems. To illustrate the promise of this approach, we share data in which future secondary mathematics teachers generated and explained *without direct instruction* sound methods for dividing by fractions and solving proportional relationships. The results are noteworthy, because these and other topics related to multiplication pose perennial challenges for many teachers.