# Lesson 5

More Division

## Warm-up: Estimation Exploration: Large Quotient (10 minutes)

### Narrative

The purpose of an Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information.

### Launch

• Groups of 2
• Display the expression.
• “What is an estimate that’s too high? Too low? About right?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

$$9,\!953\div37$$

Record an estimate that is:

too low about right too high
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### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How do you know 100 is too low?” (Because $$100 \times 37 = 3,\!700$$ and that’s less than 9,953.)
• “How can you use the value of the product $$100 \times 37$$ to estimate the value of $$9,\!953 \div 37$$?” (I know that 9,953 is more than $$2 \times 3,\!700$$ but less than $$3 \times 3,\!700$$ so the quotient is more than 200 but less than 300.)

## Activity 1: Elena’s Work (20 minutes)

### Narrative

The purpose of this activity is for students to revisit the partial quotients method to find whole number quotients. Students compare their strategy with Elena's strategy and reason about the similarities and differences using their understanding of place value. They may use estimation to identify that Elena's answer is not reasonable while they may also use parts of their own calculation to identify Elena's error (MP3).

### Launch

• Groups of 2
• “Complete the first problem.”
• 1–2 minutes: independent work time

### Activity

• “Work with your partner to complete the second, third, and fourth problems.”
• 5–7 minutes: partner work time
• “Now you will have a chance to revisit your work from the first problem.”
• 1–2 minutes: independent work time
• Monitor for students who:
• revised their original solution
• used different partial quotients

### Student Facing

1. Find the value of the quotient.

2. Here is how Elena found the quotient. Is her answer reasonable?

3. What parts of the work do you agree with? Be prepared to explain your reasoning.
4. What parts of the work do you disagree with? Be prepared to explain your reasoning.
5. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share different partial products.
• Keep student work displayed or display provided solutions.
• “How are the solutions the same? How are they different?” (They both subtract some groups of 100, 10, and 1. They both show that the value of the quotient is 521. The first strategy takes out all the hundreds at once while the second strategy takes out 400 and then 100 more.)
• “Why are multiples of 100 good to subtract?” (I can calculate them quickly and they make the number I’m dividing get smaller quickly.)

## Activity 2: Partial Quotients Practice (15 minutes)

### Narrative

The purpose of this activity is for students to practice using partial quotients. Students compare their strategy with the strategies of their classmates. They reason about the similarities and differences using their understanding of place value, balancing the complexity of calculations versus subtracting a large amount quickly.

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Engagement: Develop Effort and Persistence. Students may benefit from feedback that emphasizes effort, and time on task. For example, invite students who used one method to use a different method and compare the time it took for each approach.
Supports accessibility for: Conceptual Processing, Social-Emotional Functioning

### Launch

• Groups of 2, then 4
• “You and your partner will each find a quotient independently. After you’re done, discuss your work with your partner.”

### Activity

• 3–5 minutes: independent work time
• 1–3 minutes: partner discussion
• “Now, find another group of 2 and compare your work. How is it the same? How is it different?”

### Student Facing

1. Use partial quotients to find the value of one of the quotients. Be prepared to explain how you found the quotient.

Partner A:

Partner B:

2. Explain to your partner how you found the quotient in your problem.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask selected students who used different partial quotients to share or display or write work as shown below for all to see.
• Display:
• “How are the calculations the same? How are they different?” (They both show that the quotient is 71. One of them finishes very quickly because it starts by taking out 70 groups of 32. The other one takes longer because it subtracts smaller groups of 32.)
• “Which calculation do you prefer?” (The one that takes out all of the thirty-twos at once because it is fast. The one that takes more steps because each calculation is easier to do in my head and I know that I can subtract each multiple of 32, that is I am not trying to take out too much.)

## Lesson Synthesis

### Lesson Synthesis

“Today we compared different ways to find whole number quotients. What questions do you still have about finding whole number quotients?” (Will our method work with larger numbers? Do you always get the same answer no matter which groups you choose to subtract? Is there another way so I don't have to choose the groups at each step?)

## Cool-down: Partial Quotients (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

We investigated some different ways to find products and quotients, making sure to estimate the value before calculating. For example, the product $$49 \times 68$$ is about $$50 \times 70$$ or $$3,\!500$$. We looked at two different ways to show the newly composed units.

We also found quotients using partial products and saw that there are many different ways to do this.

The first calculation uses only 2 products but the products are more challenging to calculate. The second calculation uses 4 products but they are easier to calculate.