# Lesson 6

What’s the Quotient?

## Warm-up: Number Talk: Divide by 3 and by 6 (10 minutes)

### Narrative

This Number Talk encourages students to look for and use the structure of base-ten numbers and properties of operations to mentally find the value of division expression (MP7). The reasoning elicited here will be helpful later in the lesson when students find quotients of multi-digit numbers.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$48 \div 3$$
• $$480 \div 3$$
• $$528 \div 3$$
• $$5,\!280 \div 3$$

### Activity Synthesis

• “How is each expression related to the one before it?”
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Unfinished Divisions (15 minutes)

### Narrative

In a previous lessons, students saw that there are many ways to find products of multi-digit numbers. In this activity, students analyze and connect different ways to divide a multi-digit whole number by a single-digit whole number, and complete calculations to find the value of the quotient. In the synthesis, students compare the different methods and explain their preference.

Engagement: Provide Access by Recruiting Interest. Provide choice. Tell students they will be finding the value of $$7,\!465 \div 5$$, and that there are four unfinished strategies to look at. Invite students to choose whether they want to solve it in their own way or look at the unfinished strategies first.
Supports accessibility for: Organization, Attention, Social-Emotional Functioning

### Launch

• Groups of 2–4
• “Choose at least two calculations to finish. Make sure each calculation is completed by someone in your group.”

### Activity

• 3–4 minutes: independent work time
• 2 minutes: small-group discussion

### Student Facing

Here are four calculations to find the value of $$7,\!465 \div 5$$, but each one is unfinished.

Complete at least two of the unfinished calculations. Be prepared to explain how you know what to do to complete the work.

\displaystyle \begin{align} 5,\!000 \div 5 &= 1,\!000\\ 60 \div 5 &=\phantom{0,\!0}12\\ 5 \div 5 &=\phantom{0,\!00}1 \end{align}

7,465 is a little less than 7,500.

\begin{align}7,\!500 \div 5 &= 1,\!500\\ 35 \div 5 &=\phantom{0,\!00}7 \end{align}

### Student Response

Students may determine quotients other than 1,493. Consider asking:

• “How did you make sense of this method? How would you explain the numbers in it?”
• “How is this method like the others? How is it different?”
• “How could you use multiplication to check the value of the quotient you found?”

### Activity Synthesis

• “How are the four strategies the same? How are they different?” (The first three are the same because they involve partial quotients. Each one records the partial quotients in different ways. The last one involves estimation.)
• “Which method or methods do you find easy to follow? Which did you find hard to follow?”
• “Which method uses the most steps? Which uses the fewest steps?”
• “Which methods would work to find the value of any quotient? Which might work for this expression, but might be less useful for others?”

## Activity 2: Where Do We Begin? (20 minutes)

### Narrative

This activity serves two goals. First, it prompts students to consider whether the order in which parts of the dividend are divided makes a difference in the process or in the result. Second, it deepens students’ understanding of the structure of algorithms that use partial quotients.

Students first explain why different initial steps could be equally productive for starting a division process. Next, they analyze and complete some partial-quotients calculations with missing numbers. The missing numbers could be partial quotients, parts of the dividend being removed, or results of subtraction. To find the unknown numbers, students need to recognize and make use of the structure of the algorithm (MP7). Lastly, students use the algorithm to find a quotient, being mindful of their starting move and of the efficiency of their process.

MLR2 Collect and Display. Collect the language students use to explain how they found the quotient. Display words and phrases such as: “quotient,” “partial quotient,” and “dividend.” During the synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

• Groups of 2

### Activity

• 6–8 minutes: independent work time on the first two sets of questions
• 2–3 minutes: partner discussion
• Monitor for students who:
• can clearly explain why Jada and Noah’s initial steps are both effective
• recognize the structure of the partial quotients method and can articulate how it helps to find the missing numbers
• Pause for a discussion before the last question. Select students to share responses and reasoning.
• When discussing the second set of questions, ask: “How do you determine what the missing numbers were?” Display the incomplete calculations to facilitate students’ explanations.
• Consider annotating the calculations to clarify the structure (for instance, by drawing arrows between partial quotients and the corresponding parts of the dividend being subtracted, labeling the parts, and so on).
• 3–4 minutes: independent work time on the last question
• Monitor for students who take different first steps to divide 5,016 by 8.

### Student Facing

1. Jada and Noah are finding the value of $$3,\!681 \div 9$$. Jada says to start by dividing 81 by 9. Noah says start by dividing 3,600 by 9.

1. Explain why each suggestion is helpful for finding the quotient.
2. Find the value of $$3,\!681 \div 9$$. Show your reasoning.

2. Find the missing numbers such that each calculation shows a correct division calculation.

3. Consider the expression $$5,\!016 \div 8$$.

1. What would you do to start finding the value of the quotient?
2. Show how you would find the value with as few steps as possible.

### Student Response

Students may find some, but not all of the missing numbers in the algorithms. Consider asking:

• “Which missing numbers are you sure are accurate? How do you know?”
• “Could you use multiplication to find the missing numbers? How might that work?”
• “Could you work backwards to find the missing numbers? How would that help?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we studied different ways to divide multi-digit numbers and single-digit divisors.“

Select students who took different initial steps to find $$5,\!016 \div 8$$ to share their calculations. Discuss: