Lesson 7
Finding an Algorithm for Dividing Fractions
7.1: Multiplying Fractions (5 minutes)
Warmup
This warmup 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.
Launch
Give students 2–3 minutes of quiet work time to complete the questions. Ask them to be prepared to explain their reasoning.
Student Facing
Evaluate each expression.
 \(\frac 23 \boldcdot 27\)
 \(\frac 12 \boldcdot \frac 23\)
 \(\frac 29 \boldcdot \frac 35\)
 \(\frac {27}{100} \boldcdot \frac {200}{9}\)
 \(\left( 1\frac 34 \right) \boldcdot \frac 57\)
Student Response
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Activity Synthesis
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.
7.2: Dividing by Nonunit Fractions (15 minutes)
Activity
Here students continue to use tape diagrams to reason about division and extend their observations about unit fractions to nonunit 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.
Launch
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.
Student Facing
 To find the value of \(6 \div \frac 23\), Elena started by drawing a diagram the same way she did for \(6 \div \frac 13\).
 Complete the diagram to show how many \(\frac 23\)s are in 6.
 Elena says, “To find \(6 \div \frac23\), I can just take the value of \(6 \div \frac13\) and then either multiply it by \(\frac 12\) or divide it by 2.” Do you agree with her? Explain your reasoning.

For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression. Think about how you could find that value without counting all the pieces in your diagram.

\(6 \div \frac 34\)Value of the expression:___________

\(6 \div \frac 43\)Value of the expression:___________

\(6 \div \frac 46\)Value of the expression:___________


Elena examined her diagrams and noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:
“My first step was to divide each 1 whole into as many parts as the number in the denominator. So if the expression is \(6 \div \frac 34\), I would break each 1 whole into 4 parts. Now I have 4 times as many parts.
My second step was to put a certain number of those parts into one group, and that number is the numerator of the divisor. So if the fraction is \(\frac34\), I would put 3 of the \(\frac 14\)s into one group. Then I could tell how many \(\frac 34\)s are in 6.”
Which expression represents how many \(\frac 34\)s Elena would have after these two steps? Be prepared to explain your reasoning.
 \(6 \div 4 \boldcdot 3\)
 \(6 \div 4 \div 3\)
 \(6 \boldcdot 4 \div 3\)
 \(6 \boldcdot 4 \boldcdot 3\)

Use the pattern Elena noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.
 \(6 \div \frac27\)
 \(6\div\frac{3}{10}\)
 \(6 \div \frac {6}{25}\)
Student Response
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Student Facing
Are you ready for more?
Find the missing value.
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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 \(\frac {1}{a}\)).
Design Principle(s): Support sensemaking, Cultivate conversation
7.3: Dividing a Fraction by a Fraction (15 minutes)
Activity
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 nonunit 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.
Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their 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.
Students using the digital materials can use an applet to investigate division of fractions.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Cultivate conversation; Support sensemaking
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”

Partner A:
Find the value of each expression by completing the diagram.

\(\frac 34 \div \frac 18\)
How many \(\frac 18\)s in \(\frac 34\)?

\(\frac {9}{10} \div \frac 35\)
How many \(\frac 35\)s in \(\frac{9}{10}\)?
Use the applet to confirm your answers and explore your own examples.
Partner B:
Elena said: “If I want to divide 4 by \(\frac 25\), I can multiply 4 by 5 and then divide it by 2 or multiply it by \(\frac 12\).”
Find the value of each expression using the strategy Elena described.
 \(\frac 34 \div \frac 18\)
 \(\frac{9}{10} \div \frac35\)


What do you notice about the diagrams and expressions? Discuss with your partner.

Complete this sentence based on what you noticed:
To divide a number \(n\) by a fraction \(\frac {a}{b}\), we can multiply \(n\) by ________ and then divide the product by ________.

Select all equations that represent the statement you completed.
 \(n \div \frac {a}{b} = n \boldcdot b \div a\)
 \(n \div \frac {a}{b}= n \boldcdot a \div b\)
 \(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
 \(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
Student Response
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Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their 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.
Students using the digital materials can use an applet to investigate division of fractions.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Cultivate conversation; Support sensemaking
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”

Partner A: Find the value of each expression by completing the diagram.

\(\frac 34 \div \frac 18\)
How many \(\frac 18\)s in \(\frac 34\)?

\(\frac {9}{10} \div \frac 35\)
How many \(\frac 35\)s in \(\frac{9}{10}\)?
Partner B:
Elena said, “If I want to divide 4 by \(\frac 25\), I can multiply 4 by 5 and then divide it by 2 or multiply it by \(\frac 12\).”
Find the value of each expression using the strategy Elena described.

\(\frac 34 \div \frac 18\)

\(\frac{9}{10} \div \frac35\)


What do you notice about the diagrams and expressions? Discuss with your partner.

Complete this sentence based on what you noticed:
To divide a number \(n\) by a fraction \(\frac {a}{b}\), we can multiply \(n\) by ________ and then divide the product by ________.

Select all the equations that represent the sentence you completed.
 \(n \div \frac {a}{b} = n \boldcdot b \div a\)
 \(n \div \frac {a}{b}= n \boldcdot a \div b\)
 \(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
 \(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
Student Response
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Activity Synthesis
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.
Lesson Synthesis
Lesson Synthesis
In this lesson, we noticed a moreefficient way to divide fractions. We found that to divide \(\frac32\) by \(\frac 25\), for example, we can multiply \(\frac 32\) by 5 and then by \(\frac 12\), or simply multiply \(\frac32\) by \(\frac52\).
Let's see how this is the same or different than finding the quotient using tape diagrams. (If time permits, consider illustrating each diagram for all to see.)
 “Suppose we interpret \(\frac32 \div \frac 25\) to mean ‘how many \(\frac 25\) are in \(\frac 32\)?’ and use a tape diagram to find the answer. Where do we see the multiplication by 5 and by \(\frac 12\) in the diagramming process?” (We draw a diagram to represent \(\frac 32\) and draw equal parts, each with a value of \(\frac15\). We count how many groups of \(\frac25\) there are. Partitioning into fifths gives as 5 times as many parts. This is the multiplication by 5. Counting by twofifths leads to half as many parts. This is the multiplication by \(\frac12\).)
 “Suppose we interpret \(\frac32 \div \frac 25\) to mean ‘\(\frac 25\) of what number is \(\frac32\)?’ and use a tape diagram to find the answer. Where do we see the multiplication by 5 and by \(\frac 12\) in the diagramming process?” (We draw a tape diagram to represent a whole group. We mark twofifths of it as having a value of \(\frac 32\). We divide that value by 2 (or multiply by \(\frac12\)) to find one fifth of a group. To find out how much is in the whole group, we multiply by 5.)
Note that in both cases, there is a multiplication by \(\frac 12\) and another multiplication by 5, which is the same as multiplication by \(\frac 52\). Highlight that dividing by \(\frac ab\) is equivalent to multiplying by \(b\) and then by \(\frac 1a\), or simply multiplying by \(\frac ba\) (the reciprocal of \(\frac ab\)). This is true whether we interpreted the division problem in terms of finding the number of groups or finding the size of a group.
7.4: Cooldown  Watering A Fraction of House Plants (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
To answer the question “How many \(\frac 13\)s are in 4?” or “What is \(4 \div \frac 13\)?”, we can reason that there are 3 thirds in 1, so there are \((4\boldcdot 3)\) thirds in 4.
In other words, dividing 4 by \(\frac13\) has the same result as multiplying 4 by 3.
\(\displaystyle 4\div \frac13 = 4 \boldcdot 3\)
In general, dividing a number by a unit fraction \(\frac{1}{b}\) is the same as multiplying the number by \(b\), which is the reciprocal of \(\frac{1}{b}\).
How can we reason about \(4 \div \frac23\)?
We already know that there are \((4\boldcdot 3)\) or 12 groups of \(\frac 13\)s in 4. To find how many \(\frac23\)s are in 4, we need to put together every 2 of the \(\frac13\)s into a group. Doing this results in half as many groups, which is 6 groups. In other words:
\(\displaystyle 4 \div \frac23 = (4 \boldcdot 3) \div 2\)
or
\(\displaystyle 4 \div \frac23 = (4 \boldcdot 3) \boldcdot \frac 12\)
In general, dividing a number by a fraction \(\frac{a}{b}\) is the same as multiplying the number by \(\frac{b}{a}\), which is the reciprocal of the fraction.