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Dividing Into Groups of Fractions

If you have considered the ideas we shared in the videos Divide into Groups and Fraction Division Two Ways, then you have begun to construct an understanding of division as a multiplicative idea. You may also have come to recognize that the same conception of division can be applied to dividing fractions. We don't need a new procedure. We need only read and reason with the division as dividing the dividend into groups of the divisor. (You may remember that 8 ÷ 4 can be read as "eight divided into groups of 4." Then, the quotient (2) is the number of groups.) In this video, Eric shares one way of reasoning with fraction division that offers a conceptual foundation for understanding the commonly taught algorithm. Here, we see one way to reason with fraction division is to have the fractions be counts of the same size pieces. That is, it works to have like fractions, with common denominators. Consider dividing ⅘ into groups of ⅖. Each group has ⅖ in it, so there would be 2 groups. But, the fractions in this video aren't like fractions. Eric creates like fractions much like Kaylie reasoned with fifths and tenths in this video on addition. With the first problem, Eric engages in relational thinking starting with the fact that 2(½) = 1 to arrive at the equivalence of ½ and 4(⅛). He could have also started with 8(⅛) = 1 and halved the number of eighths to yield 4(⅛) = ½. When you are Learning for the Long-View, you want to explore mathematically sound alternatives before arriving at the most efficient route. This allows you to grow your strength, stamina, and flexibility in reasoning with numbers. We are shooting for high quality thinking. As Eric considers the second problem, he reasons with creating like fractions with thirds and fourths. Again, rather than invoke a memorized procedure, Eric guides you to consider how you can reason that ¼· ⅓ = ¹⁄₁₂ in which case 4(¼)· ⅓, which is one whole ⅓, is equivalent to ⁴⁄₁₂. He uses an area model along with intentional language to help guide the reasoning and reassures us that in time we will internalize the area model. Then, the reasoning will be guided by intentional language. We recognize that language is key to constructing knowledge. At Long-View, we make a consistent effort to use language that supports reasoning. This article is helpful as you begin to practice using language that supports mathematical reasoning. You might also find this quick reference on language helpful.