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"Would you rather have half of one chocolate bar or a quarter of a different chocolate bar? Most popular answer: It depends on the size of the chocolate bars! As teachers know, the relative size of fractions depends on how the whole is defined. Authors Jim Callahan and Marilynn Varricchio address these common problems with fractions in their new book Fractions Made Easy (Didax, 2016). Drawing on material from the book, we will focus on how visual models can be used to support a solid conceptual understanding of fractions in third grade.

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Double, or dual, number lines are an ideal "visual model" for solving mathematical problems involving equivalences, ratios, proportions, and more. Though double number lines are not mentioned specifically in the CCSS until Grade 6 (6.RP.3), they are a viable tool for mathematical understanding beginning as early as first grade.Read More
A new manipulative has emerged that appears to provide many students with a better understanding of early number concepts. The Rekenrek, arithmetic rack, or counting rack was developed at the Freudenthal Institute in the Netherlands by Adrian Treffers. Resembling an abacus, the Rekenrek typically consists of two rows of ten beads, with each row having five red beads and five white beads. The Rekenrek takes its place in the primary grades classroom alongside other popular models for developing early mathematics concepts, such as Unifix Cubes, base ten blocks, ten-frames and counters. The focus in using a Rekenrek is on fives and tens. Rekenreks are also available with four rows of beads or ten rows of beads to deal with numbers 21 through 100.Read More
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