In this section, students reason about multiplication of two fractions. They begin by considering situations that involve finding a fraction of a fraction. Students represent the situations by drawing diagrams that make sense to them.

For example, “A pan of macaroni and cheese is full. Kiran eats of the macaroni and cheese. How much of the whole pan did Kiran eat?”

By partitioning the first third of a pan into fourths and doing the same with the other two thirds, students can see that Kiran ate of the whole pan.

Diagram. Square, length and width, 1. Partitioned horizontally into 3 equal rectangles. top rectangle partitioned vertically into 4 equal rectangles. 1 rectangle shaded dark blue, 3 rectangles shaded light blue.

Students connect the product of two fractions to the area of a rectangle with fractional side lengths. When multiplying unit fractions, students see the denominator as the number of equal parts in the unit square, structured as an array. So partitioning one side of a rectangle into fourths and the other into thirds creates a 4-by-3 array. Each part in the array is of 1 whole.

 The area of a rectangle that is by is , or . Students generalize this as:

They extend this insight to find the product of non-unit fractions, including fractions greater than 1.

For example, the value of  is because parts are shaded and there are equal parts in 1 whole.

In this section, students solve problems involving multiplication and division of fractions. As they reason about situations and interpret tape diagrams, students see that the same situation or diagram can be expressed with multiplication or division.

For example, if gallon of lemonade is shared equally by 5 friends, each friend gets  gallon of lemonade. This also means that each friend gets of the gallon, which can be expressed by .

Students interpret situations and diagrams in terms of one or both operations, depending on what makes sense in the given context. In this diagram, the shaded part represents both and .

In IM Grade 3, students learned that division can be understood in terms of equal-size groups and can be interpreted in two ways. For example, can mean finding the size of each group if 8 is put into 4 equal groups, or finding how many groups of 4 are in 8.

In this section, students extend this idea to divide a unit fraction by a whole number and divide a whole number by a unit fraction. They interpret to mean finding the size of one part if is split into 5 equal parts, and as a way of finding how many s are in 5.

To build this understanding, students reason about situations, diagrams, and expressions that represent division. Students look for patterns and assess the reasonableness of the quotients they find.

Students may notice that to find , they can multiply 5 by 2 because there are 2 halves in each of the 5 wholes. It is not essential that students generalize division of fractions at this point, as they will do so in IM Grade 6.