This unit begins with a review of fraction arithmetic from fifth grade—specifically, addition, subtraction, simplification, and multiplication of fractions. Then it focuses on the new topic: division of fractions.
The introductory lesson on the division of fractions presents the concept of reciprocal numbers and ties the reciprocity relationship to the idea that division is the appropriate operation to solve questions of the form, “How many times does this number fit into that number?” For example, we can write a division from the question, “How many times does 1/3 fit into 1?” The answer is, obviously, 3 times. So we can write the division 1 ÷ (1/3) = 3 and the multiplication 3 × (1/3) = 1. These two numbers, 3/1 and 1/3, are reciprocal numbers because their product is 1.
Students learn to solve questions like that through using visual models and writing division sentences that match them. The eventual goal is to arrive at the shortcut for fraction division—that each division can be changed into a multiplication by taking the reciprocal of the divisor, which is often called the “invert (flip)-and-multiply” rule.
However, that “rule” is just a shortcut. It is necessary to memorize it, but memorizing a shortcut does not help students make sense conceptually out of the division of fractions—they also need to study the concept of division and use visual models to better understand the process involved.
In two lessons that follow, students apply what they have learned to solve problems involving fractions or fractional parts. A lot of the problems in these lessons are review in the sense that they involve previously learned concepts and are similar to problems students have solved earlier, but many involve the division of fractions, thus incorporating the new concept presented in this unit.