Lesson 7

This warm-up revisits multiplication of fractions from grade 5. Students will use this skill as they divide fractions throughout the lesson and the rest of the unit. 

Give students 2–3 minutes of quiet work time to complete the questions. Ask them to be prepared to explain their reasoning.

Ask a student to share their answer and reasoning to each question, then ask if anyone disagrees. Invite students who disagree to share their explanations. If not mentioned in students' explanations, discuss strategies for multiplying fractions efficiently, how to multiply fractions in which a numerator and a denominator share at least one common factor, and how to multiply mixed numbers.

If students mention "canceling" a numerator and a denominator that share a common factor, demonstrate using the term "dividing" instead. For example, if a student suggests that in the second question (\(\frac 12 \boldcdot \frac 23\)) the 2 in \(\frac 12\) and the 2 in the \(\frac 23\) "cancel out", rephrase the statement by saying that dividing the 2 in the numerator by the 2 in the denominator gives us 1, and multiplying by 1 does not change the other numerator or denominator. 

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.

This activity was originally designed to follow another activity which is not included in the sequence for this course. In the launch for this activity, display the following question and image:

To find the value of \(6 \div \frac13\), Elena thought, “How many \(\frac13\)s are in 6?” and then she drew this tape diagram. It shows 6 ones, with each one partitioned into 3 equal pieces.

Ask students to answer Elena’s question and explain their reasoning. Make sure students understand why the answer is 18 before proceeding with this activity.

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 \(\frac {1}{a}\)).

This is the final task 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 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.

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:

• Dividing by a whole number \(n\) is the same as multiplying by a unit fraction \(\frac{1}{n}\) (e.g., dividing by 5 is the same as multiplying by \(\frac15\)).

• Dividing by a unit fraction \(\frac{1}{n}\) is the same as multiplying by a whole number \(n\) (e.g., dividing by \(\frac 17\) is the same as multiplying by 7).

• Dividing by a fraction \(\frac{a}{b}\) is the same as multiplying by a unit fraction \(\frac{1}{a}\) and multiplying by a whole number \(b\), which is the same as multiplying by \(\frac ba\) (e.g., dividing by \(\frac57\) is the same as multiplying by \(\frac 15\), and then by 7. Performing these two steps gives the same result as multiplying by \(\frac75\)).

Finish the discussion by trying out the generalized method with other fractions such a \(18\div\frac97\), or \(\frac{15}{14}\div\frac{5}{2}\). Explain that although we now have a reliable and efficient method to divide any number by any fraction, sometimes it is still easier and more natural to think of the quotient in terms of a multiplication problem with a missing factor and to use diagrams to find the missing factor.