In this activity, students use tape diagrams and one interpretation of division to divide a number by unit fractions. They do this as a first step toward generalizing the reasoning for dividing any two fractions. By reasoning repeatedly and noticing a pattern (MP8), students arrive at the conclusion that is equivalent to .

As students work, monitor for those who are able to apply the reasoning in the first two problems to subsequent problems without the help of diagrams. Select several students to share later.

The goal of the discussion is to make explicit the connection between division by a unit fraction and multiplication by the whole number that is its denominator, using tape diagrams to highlight this connection.

Invite students to share their completed diagrams for , , and . (If needed, remind students that fractions such as , , and are called “unit fractions.”) Ask questions such as:

• “Where in each diagram do we see division by the unit fraction , , or ?” (The tape is broken into equal parts. The unit fraction is the size of each part.)
• “Where in each diagram do we see the multiplication of 6 by 3, 4, or 6?” (In each diagram, we see the 6 ones divided into multiple equal parts: 6 groups of 3 thirds in the first diagram, 6 groups of fourths in the second, and 6 groups of sixths in the third.)
• “How many parts would be on a tape diagram that represents 6 divided by ?” (60 tenths) “What about 6 divided by ?” (300 fiftieths)

Next, ask previously selected students to share their responses to the last set of questions. Emphasize that when we divide a number by a unit fraction , we end up with times as many parts, so dividing by is the same as multiplying by .

In this activity, students extend their reasoning about division by unit fractions to division by non-unit fractions. Specifically, they explore how to represent the numerator of the fraction in the tape diagram and study its effect on the quotient. Students generalize their observations as operational steps and then as an expression. Along the way, they practice looking for and making use of structure (MP7).

As students work, monitor for those who effectively show the fractional divisor on their diagrams, as well as those who could explain why the steps make sense. Select them to share their diagrams or reasoning later.

Focus the discussion on clarifying why Elena's two steps make sense and the expression that represents her process of finding .

Invite previously selected students to share their tape diagram, explanation, and expression for finding , or display the following diagrams for all to see.

Fraction bar diagram. 24 equal parts. Every fourth part has a solid line. All other lines are dashed. First part labeled the fraction 1 over 4. Total labeled 6.

Discuss questions such as:

• “Where in the diagrams can we see Elena’s first step—dividing each 1 whole into as many parts as the number in the denominator?” (Each of the 6 wholes is partitioned into 4 parts.)
• “Where can we see ‘4 times as many parts?’” (Instead of having 6 wholes, there are now 24 fourths.)
• “Where can we see Elena’s second step—putting a certain number of the parts into groups and seeing how many groups there are?” (Every 3 one-fourths make a group, so there are 8 groups in 24 fourths.)
• “Why does make sense?” (We multiplied 6 by 4 to get the number of fourths, and then divided it by 3  to get the number of groups containing 3 one-fourths.)