Lesson 10

In this warm-up, students use tape diagrams to revisit the idea that dividing by a whole number is equivalent to multiplying by a unit fraction. Though this is a review of a grade 5 expectation, connecting the division problems to diagrams allows students to see the equivalence in the related division and multiplication problems. It also prepares students to extend the reasoning and representations used here to division by non-unit fractions later. 

Arrange students in groups of 2. Ask one person in each group to draw diagrams and answer the questions for Partner A, and the other to take on the questions for Partner B. Give students a few minutes of quiet time to complete the first two questions, and then ask them to collaborate on the last two.

Invite a couple of students to share their observations about their group's diagrams and answers. Students should notice that the answers for the three division problems match those for the multiplication ones, even though the questions were not the same and their diagrams show groups of different sizes. Ask a few students to share their response to the last question.

Consider displaying the following image to reinforce the idea that dividing by a whole number has the same effect as multiplying by the reciprocal of that number.

In this activity, students use tape diagrams and the meanings 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 \(a \div \frac1b\) is equivalent to \(a \boldcdot b\).

Later, students put together those results to see that dividing by a fraction \(\frac{c}{d}\) is equivalent to dividing by \(c\) and multiplying by \(d\).

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

Arrange students in groups of 2. Give students 5–6 minutes of quiet work time and then time to share their work with a partner. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups on a tape diagram by coloring.

For classes using the digital materials, an applet from an earlier lesson can also be used to represent the division problems here,

For some students, the structure in the division may still not be apparent by the time they get to problems such as \(6 \div \frac{1}{25}\), and they may try to partition a section in a tape diagram into 25 parts. Ask them to study the diagrams as well as the answers to the previous questions carefully and to look for a pattern.

Select previously identified students to share their responses to the questions \(6 \div \frac {1}{b}\) and \(a \div \frac {1}{b}\). To highlight the connections between the diagram, division by a unit fraction, and multiplication by the reciprocal of the fraction ask students:

• “How is the division by a unit fraction depicted in the diagrams?” (The tape diagram is broken into equal parts. The unit fraction is the size of each part.)
• “Where in the diagrams do we see the multiplication?” (The multiplication shows the number of parts in each 1 whole.)
• “How are the two—the division by a unit fraction and the multiplication—related?” (When we divide a number by a unit fraction \(\frac1b\), we end up with \(b\) times as many parts, so dividing by \(\frac1b\) is the same as multiplying by \(b\).)

If not already articulated by students, clarify that dividing a number by a unit fraction has the same result as multiplying by its reciprocal.

Discuss the usefulness and limits of diagrams. Ask:

• “How do we find the value of \(1,\!000 \div \frac 19\) or \(2 \div \frac{1}{250}\) using a diagram?” (By dividing each 1 whole into 9 parts or 250 parts)
• “Would you use diagrams to find these quotients? Why or why not?” (No, it is not practical.)
• “When working on the task, did you stop partitioning the tape diagrams at some point? If so, why?”
• “Why do we use diagrams? When can they be helpful?” (Diagrams can show us the structure or relationships between numbers and help us see the general process.)

If not mentioned by students, point out that for larger numbers, or smaller fractions, drawing a full diagram becomes increasingly cumbersome. Noticing the structure that is visible in the diagrams for easier cases allows us to use it for more difficult ones.

Here students continue to use tape diagrams to reason about division and extend their observations about unit fractions to 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 expressions, which they then use to solve other division problems. 

As students work, notice those who effectively show the divisor on their diagrams, i.e., the multiplication by the denominator and division by the numerator, as well as those who could explain why the steps make sense.

Keep students in groups of 2. Give students 5–7 minutes of quiet work time and then time to share their responses with their partner. Provide continued access to colored pencils.

Some students may read the phrase “partition 1 section into 4 parts” in Elena's reasoning and focus on the value of each small part \(\left(\frac 14\right)\) instead on how many parts are now shown. Similarly, they may take “making of 3 of these parts into one piece” to imply multiplying the \(\frac14\) by 3, instead of looking at how it changes the number of pieces. Explain that Elena's diagram& suggests that she interpreted \(6 \div \frac34\) as “how many \(\frac34\)s are in 6?” which tells us that we are looking for the number of groups, rather than the value of each part in the diagram.

Select a few students to share their diagrams  and explanations about why Elena's reasoning and method work. Display the following tape diagram for \(6 \div \frac 34\), if needed. Ask students to point out where in the diagram the two steps are visible. 

To involve more students in the conversation, consider asking questions such as:

• “Who can restate ___'s reasoning in different words?”
• “Did anyone think about the division the same way but would explain it differently?”
• “Does anyone want to add an observation to the way ____ reasoned about the division?”
• “Do you agree or disagree? Why?”

Highlight that dividing a number \(c\) by a fraction \(\frac {a}{b}\) has the same result as multiplying by \(b\), then dividing by \(a\) (or multiplying by \(\frac {1}{a}\)).