In this section, students develop a general algorithm for dividing a fraction by a fraction, pulling together the thread of reasoning from earlier lessons.

First, students use tape diagrams to reason repeatedly about division of a whole number by a unit fraction. they look for structure and regularity in the process. Then, they do the same with division by a non-unit fraction. Students generalize that to divide a number \(n\) by a non-unit fraction \(\frac ab\)is to figure how many unit fractions \(\frac1b\)are in \(n\) (find \(n \div \frac1b\)) and then see how many groups of \(a\) unit fractions can be formed (find \(n \div \frac 1b \div a\)).

For example, to divide 6 by \(\frac 34\) is to figure how many \(\frac 14\)s are in 6 (find \(6 \div \frac 14\)) and see how many groups of three \(\frac 14\)s can be formed (find \(6 \div \frac 14 \div 3\)).

Next, students extend their generalized steps to divide a fraction by a fraction. they conclude that to find \(n \div \frac ab\) is to find \(n\cdot{b}\cdot\frac1a\), or \(n\cdot\frac ba\). Finally, students practice applying this algorithm and using it strategically to calculate quotients.

6.4 Section C: Algorithm for Fraction Division