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• Author or Editor: Xiaowen Cui
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## Collaboratively Engaging with GCFs and LCMs

Collaborative engagement provides an opportunity for students to construct and solidify their own knowledge and understanding of important mathematical ideas. According to Van de Walle, Karp, and Bay-Williams, “learning is enhanced when the learner is engaged with others working on the same idea” (2015, p. 52). In allowing students to work with their peers to practice problems and construct important mathematical connections, the students build on their combined prior knowledge to formulate newfound ideas and conjectures. We recognize that grouping students so that each group will function in a productive manner can often be difficult. Therefore, we have devised this activity that allows students to work together and communicate with ten different students individually. In a usual group setting, the students would get to work with one or two other students, but the format of this activity allows for more forms of mathematics communication and collaboration.

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## Abacus Interactive

Calculators are often efficient in finding the answer to an addition or subtraction problem, but they do not reveal the process by which the answer is obtained. Developing students' fluency in addition and subtraction using strategies and algorithms based on place value, composing and decomposing numbers in base 10, and reading and writing numerals in expanded form are important teaching and learning standards not only for the elementary grades but for middle school students as well (NCTM 2000; CCSSI 2010; TEA 2015). We introduced the Chinese abacus to our students as a hands-on tool to illuminate the meaning of a number in expanded form in terms of place values and to strengthen students' conceptual understanding of the standard algorithms of addition and subtraction. “Students' understanding of the base 10 number system is deepened as they come to understand its multiplicative structure” (NCTM 2000, p. 143). This activity will let students explore the mathematical properties of the base 10 system in a creative and interactive way. Students develop a deeper meaning of why the standard algorithms work and how they relate to a number in expanded form. This activity is best suited for elementary and middle grades.