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.)

This is the final activity in a series that leads students toward a general procedure for dividing fractions. Students verify previous observations about the steps for dividing non-unit fractions (namely, multiplying by the denominator and dividing by the numerator) and contrast the results with those found using tape diagrams. They then generalize these steps as an algorithm and apply it to answer other division questions.

As students discuss in their groups, listen to their observations and explanations. Select students with clear explanations to share later.

In the digital version of the activity, students can choose to use an applet to represent division of two fractions. The applet allows students to represent the fractions on two tape diagrams and measure how many times one fraction would fit in another. Students who need support in processing abstract visual information may benefit from the ability to represent the two fractions separately and partition each tape dynamically. If students don't have individual access, displaying the applet for all to see would be helpful.

Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout. Remind students that in the past few activities, we learned that:

Tell students that pairs of numbers such as 5 and ​​​​​,  and 7, and ​​​​ and ​​​​ are called reciprocals, or numbers that when multiplied equal 1. 

We can also say that 5 is a reciprocal of and is a reciprocal of 5. Ask students to name the reciprocal of a few other examples, such as 9, , , and

If time permits, ask students to use the generalized method to divide other fractions, such as or .