In this section, students make sense of division situations involving fractions, gradually building an understanding that can later be generalized into an algorithm.

The first four lessons explore “How many groups?” questions in various contexts. In the first lesson, the size of a group is a fraction and the number of groups is a whole number. This lesson is optional because it reinforces grade 5 work on multiplication and division of whole numbers by unit fractions.

Then, students investigate situations in which the number of groups is a fraction greater than 1 or less than 1. For the latter, it makes sense to ask “What fraction of a group?” instead of “How many groups?” To represent and reason about different situations, students use tape diagrams and equations. For instance, "How many \(\frac 35\)s are in 2?" can be represented in these ways:

\(? * \frac 35 = 2\)

\(2 \div \frac35 = ?\)

Next, student explore cases in which the size of 1 group is unknown. They learn that questions such as "If there are 3 pounds in \(2\frac14\)bags, how much is in 1 bag?" and "\(\frac 25\)of what number is 4?" can also be answered by dividing and can be represented with diagrams and equations.

The section concludes with an optional lesson to practice finding the amount in 1 group and solving various division problems in context.

6.4 Section B: Meanings of Fraction Division