Virginia Mathematics Teacher Spring 2017

fluency based on their understanding of part-whole relationships.

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Figure 1: Rekenrek

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often used to develop strategies for basic facts such as working with the five-structure, making ten, one more or less than, compensation and stretching children toward using these strategies with the support of or in place of counting (Fosnot & Dolk, 2001). The five-structure of a rekenrek offers visual support to quickly see the quantity of five without counting. This is a form of subitizing, the ability to recognize the number of objects in a set without actually counting them (Clements, 1999; MacDonald & Shuway, 2016). Subitizing with a rekenrek is a concrete representation to flexibly decomposing and composing numbers based on part-whole relationships. For example, on a rekenrek, students can see that seven is composed of five (five red beads) and two (two white beads) and eight is composed of five (five red beads) and three (three white beads). Figure 2 shows a

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Figure 3: 7+8, 5+5+2+3, and 10+5 using part-whole bar models

Part-whole bar models are a visual representation that show the magnitude of a number as well as the relationships among the whole number and its parts. In Singapore, the bar model is used as a tool for problem solving because it communicates graphically complex relationships, such as comparisons, part-whole calculations, ratios, proportions, and rates of change (Hoven & Garelick, 2007). For example, in a bar model, 15 can be decomposed visually into its parts and represented proportionally. Figure 3 shows part- whole bar models representing 7+8, 5+5+2+3, and 10+5. As students visually represent their rekenrek work with part-whole bar models and then symbolically represent the models with number sentences, they can connect concrete, representational, and abstract models of basic addition facts. These three levels of representation also provide students with an opportunity to begin an exploration of the concept of equality. CRA in Action In the classroom vignette that follows, Mr. Dominguez, a first-grade teacher, is working with students using rekenreks along with part-whole bar models to build fluency of basic addition facts based on number sense (Virginia Mathematics Standards of Learning (SOL) 1.7) and to explore the concept of equality (SOL 1.15). Mr. Dominguez’s class regularly uses manipulatives to represent and describe their thinking. In a previous

Figure 2 : 7 + 8 on the Rekenrek

rekenrek representing 7 + 8. Students may subitize and see 7 + 8 is the same as 5 + 5 + 2 + 3 or 10 + 2 + 3 or 10 + 5. Students can also see a concrete representations of the commutative property of addition (7 + 8 = 8 + 7). Using rekenreks, students can purposefully practice building combinations of numbers and develop

Virginia Mathematics Teacher vol. 43, no. 2

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