6.4.10 - Dividing by Unit and Non-Unit Fractions

Student Facing Goals I can divide a number by a non-unit fraction `\frac{a}{b}` by reasoning with the numerator and denominator, which are whole numbers. I can divide a number by a unit fraction `\frac{1}{b}` by reasoning with the denominator, which is a whole number. Lesson Narrative This is the first of two lessons in which students pull together the threads of reasoning from the previous six lessons to develop a general algorithm for dividing fractions. Students start by recalling the idea from grade 5 that dividing by a unit fraction has the same outcome as multiplying by the reciprocal of that unit fraction. They use tape diagrams to verify this. Next, they use the same diagrams to look at the effects of dividing by non-unit fractions. Through repetition, they notice a pattern in the steps of their reasoning (MP8) and structure in the visual representation of these steps (MP7). Students see that division by a non-unit fraction can be thought of as having two steps: dividing by the unit fraction, and then dividing the result by the numerator of the fraction. In other words, to divide by `\frac{2}{5}` is equivalent to dividing by `\frac{1}{5}`, and then again by 2. Because dividing by a unit fraction  is equivalent to multiplying by 5, we can evaluate division by `\frac{2}{5}` by multiplying by 5 and dividing by 2.

By Jennifer Meikrantz