# Lesson 18

Divide with Partial Quotients

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

### Narrative

This Number Talk encourages students to look for and make use of the structure of numbers in base-ten to mentally solve division problems. The reasoning elicited here will be helpful later in the lesson when students divide large numbers using increasingly more abstract strategies.

### Launch

• Groups of 2
• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$90 \div 3$$
• $$96 \div 3$$
• $$960 \div 3$$
• $$954 \div 3$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Who can restate _______ 's reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to____’s strategy?”

## Activity 1: Decompose Dividends (20 minutes)

### Narrative

In this activity, students encounter a way to divide a multi-digit number by using partial quotients and writing equations for them. They analyze and interpret the equations and consider how it is like and unlike finding quotients using base-ten representations. In the next activity, students will be introduced to a way to record partial quotients vertically.

MLR8 Discussion Supports. To support the transfer of new vocabulary to long term memory, invite students to chorally repeat the word in unison 1-2 times: partial quotient.

### Required Materials

Materials to Gather

• Groups of 4.

### Activity

• Pause after the first question and discuss students’ responses. Record and display responses for all to see.

### Student Facing

1. Find the value of $$465 \div 5$$. Show your reasoning. You may use base-blocks if you find them helpful.
2. Here’s how Priya finds the value of $$465\div 5$$.

\begin{align} 400\div 5&= 80\\ 60\div 5 &= 12\\ 5 \div 5 &= \phantom{0}1 \\ \overline {\hspace{5mm}465 \div 5} &\overline{\hspace{1mm}= 93 \phantom{000}}\end{align}

1. What has Priya done? Describe her steps.

2. How is Priya’s method similar to your method?
3. Use Priya’s method to find the value of $$428 \div 4$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their interpretations of Priya’s work and compare it to their reasoning in the first question.
• “How did Priya decompose the number 465?” (By place value. $$400 + 60 + 5$$)
• “What does Priya do after writing the first three equations?” (She adds up the quotients.)
• “We can find a quotient in parts—dividing a portion of the dividend at a time—until there is no more (or until there is not enough) of the dividend to divide.”
• “Each quotient is called a partial quotient.”

## Activity 2: Tyler’s Method (15 minutes)

### Narrative

In this activity, students are introduced to an algorithm that uses partial quotients, a vertical method of recording partial quotients. They compare and contrast this approach with other ways of dividing numbers using partial quotients and try using it to divide multi-digit numbers.

When students analyze Priya and Tyler's work and explain their reasoning, they critique the reasoning of others (MP3).

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for problem 2 before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 4
• Display Priya and Tyler’s methods.

### Activity

• “Tyler used a different method to record $$465 \div 5$$. Analyze what is happening in his method. Think about how the two methods are alike and different.”
• 3 minutes: independent work time on the first two questions
• 3 minutes: small-group discussion
• Invite students to share their analyses on the two methods.
• If not mentioned by students, highlight that both Priya and Tyler divided in parts, but reasoned and recorded differently.
• Priya recorded the partial quotients with division equations.
• The partial quotients in Tyler's work are recorded as factors being multiplied by 5, and also listed above the dividend.
• Tyler kept dividing in parts and subtracting until there's nothing left of the dividend to divide.
• Clarify the meaning of the numbers in Tyler’s method before students work on the last question.
• 3 minutes: independent work time to find the value of $$428 \div 4$$.

### Student Facing

Tyler uses a different method to find the value of $$465 \div 5$$. Let’s compare Priya’s and Tyler’s work.

Priya's method

\begin{align} 400\div 5&= 80\\ 60\div 5 &= 12\\ 5 \div 5 &= \phantom{0}1 \\ \overline {\hspace{5mm}465 \div 5} &\overline{\hspace{1mm}= 93 \phantom{000}} \end{align}

Tyler's method

1. How are Priya and Tyler’s methods alike? How are they different? List as many similarities and differences as you can find.
2. Why do you think Tyler uses subtraction in his method?
3. Show how Tyler might record the process of finding the value of $$428 \div 4$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses. Highlight the different ways to decompose 428 or the different partial quotients that could be used to find $$428 \div 4$$.
• Make sure students see that Priya’s equations and Tyler’s method are simply two ways to record partial quotients, but they are not fundamentally different.
• “Tyler’s vertical recording method is another type of algorithm.”

## Lesson Synthesis

### Lesson Synthesis

“Today we learned to use an algorithm that uses partial quotients to divide numbers.”

“How would you explain ‘partial quotients’ to a classmate who might be absent today?” (We can find a quotient in parts—dividing a portion of the dividend at a time—until there is no more or until there is not enough of the dividend to divide. Each quotient is called a partial quotient.)

“Suppose we’d like to find the value of $$738 \div 9$$ and know we could decompose the 738 into parts. How would we know what numbers to choose?” (Look for multiples of 9. Try to start with the largest multiple of 9 and 10 within 738.)

“What are some ways to decompose 738 into multiples of 9?” ($$720 + 18$$, or $$450 + 270 + 18$$, among others.)

Display:

$$738 \div 9$$

\begin{align} 720\div 9&= 80\\ 18\div 9 &= \phantom{0}2\\ \overline {\hspace{5mm}738 \div 9} &\overline{\hspace{1mm}= 82 \phantom{000}}\end{align}

“We saw two ways of recording partial quotients—by writing a series of equations and by recording the steps of division vertically. Where can we see the partial quotients in each one?”

## Cool-down: Subtract Groups (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.