Teachers have found many different ways to support students who are learning about rational numbers. Some of the most productive ways often involve the use of representations that anchor students' experiences in the quantities being learned. Although almost all representations have their limitations, they also provide opportunities to support students in pressing their understanding of rational numbers. In this article, I share an activity motivated by a discussion that occurred during one of my seventh-grade classes. We used a hundredths diagram, in which the area of the large square represented 1 unit (for other interesting uses of this diagram, see Scaptura, Suh, and Mahaffey 2007 and Cramer et al. 2009).

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### Nicholas J. Gilbertson

Context and underlying structure explain why the formula for binomial coefficients always produces an integer solution

### Nicholas J. Gilbertson and Kimberly Cervello Rogers

Can you find a triangle in which the three bases and three heights are integer values?

### Jamie L. Wernet, Kevin A. Lawrence and Nicholas J. Gilbertson

Most mathematics curricula include contextual problems, but not all these problems have the same potential as modeling tasks. The authors describe how to select tasks that give students opportunities to use modeling.

### Nicholas J. Gilbertson, Samuel Otten, Lorraine M. Males and D. Lee Clark

For many American students, high school geometry provides their only focused experience in writing proofs (Herbst 2002), and proof is often viewed as the application of recently learned theorems rather than a means of establishing and understanding the truth of general results (Soucy McCrone and Martin 2009).