Lesson 16
Reason About Quotients
Warmup: Estimation Exploration: How Many One Fifths? (10 minutes)
Narrative
Launch
 Groups of 2
 Display the expression.
 “What is an estimate that’s too high? Too low? About right?”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Record responses.
Student Facing
\(98 \div \frac{1}{5}\)
Record an estimate that is:too low  about right  too high 

\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)  \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)  \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) 
Student Response
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Activity Synthesis
 “How do you know the value of \(98 \div \frac{1}{5}\) is less than 500?" (It's less than \(100 \div \frac{1}{5}\) and that's 500)
Activity 1: Greater Than or Less Than 1 (20 minutes)
Narrative
The purpose of this activity is for students to reason about the size of quotients, involving a unit fraction and a whole number, by carefully analyzing the relative sizes of the dividend and divisor rather than finding the value of the expressions. As students work, listen for the language they use to explain why they think the value of an expression is greater than or less than 1. Highlight the language during the synthesis. When students explain to each other how they decided whether a quotient is greater than 1 or less than 1 they construct viable arguments (MP3).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.
Supports accessibility for: Attention, Conceptual Processing, Memory
Launch
 Groups of 2
Activity
 1–2 minutes: quiet think time
 5–8 minutes: partner work time
Student Facing
\(25\div\frac15\)
\(\frac17\div 25\)
\(\frac18\div 25\)
\(25\div\frac17\)
\(25\div\frac18\)
\(\frac15\div 25\)

Without finding the value of the expressions, write each expression under the correct category.
The value of the expression is less than 1
The value of the expression is greater than 1
 Explain your strategy for determining whether a quotient is less than 1 or greater than 1.
Student Response
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Activity Synthesis
 “Share your explanation for determining whether a quotient is less than or greater than 1 with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
 3–5 minutes: structured partner discussion.
 Repeat with 2–3 different partners.
 If needed, display question starters and prompts for feedback.
 “Can you give an example to help show . . . ?”
 “How can you use the words divisor and dividend in your explanation?”
 “Revise your initial draft based on the feedback you got from your partners.”
 2–3 minutes: independent work time
Activity 2: Estimate and Divide (15 minutes)
Narrative
The purpose of this activity is for students to order the quotients from the previous activity from least to greatest, without calculating. The quotients of a whole number by a unit fraction have the same dividend so students reason that the expression with the smallest unit fraction divisor represents the largest quotient. In the same way, the quotients of a unit fraction by a whole number all have the same divisor so the expression with the largest unit fraction dividend is the largest.
Launch
 Groups of 2
Activity
 5 minutes: independent work time
 5 minutes: partner discussion
 Monitor for students who:
 explain that the greatest quotient is \(25 \div \frac {1}{8}\) because it represents the largest number of pieces
 explain that the smallest quotient is \(\frac {1}{8} \div 25\) because it represents the smallest sized piece
 change their response after the partner discussion
Student Facing
\(25\div\frac15\)
\(\frac17\div 25\)
\(\frac18\div 25\)
\(25\div\frac17\)
\(25\div\frac18\)
\(\frac15\div 25\)
 Without finding the value of the expressions, put the expressions in order from least to greatest.
 Choose 2 expressions and find the value of the expressions.
Student Response
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Advancing Student Thinking
 “Will the quotient be greater than or less than the dividend?”
Activity Synthesis
 Ask previously selected students to explain their reasoning.
 “What was challenging about this activity?” (It was hard to explain without finding any values.)
 “How did reasoning about the order of the expressions help you find the value of 2 of the expressions?” (I knew the value was going to be a unit fraction [or whole number].)
Lesson Synthesis
Lesson Synthesis
Display:
\(25 \div \frac{1}{\boxed{\phantom{\frac{aaai}{aaai}}}}\)
“What do we know about the value of this expression if the number in the box is a whole number?” (It is going to be greater than 25. It is going to be a multiple of 25.)
Display:
\(\frac{1}{\boxed{\phantom{\frac{aaai}{aaai}}}}\div 25\)
“What do we know about the value of this expression if the number in the box is a whole number?” (It is going to be a unit fraction. The denominator is going to be a multiple of 25.)
Cooldown: Both Types of Problems (5 minutes)
CoolDown
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