Lesson 5
More Division
Warmup: Estimation Exploration: Large Quotient (10 minutes)
Narrative
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
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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
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

Find the value of the quotient.

Here is how Elena found the quotient. Is her answer reasonable?
Explain or show your reasoning.
 What parts of the work do you agree with? Be prepared to explain your reasoning.
 What parts of the work do you disagree with? Be prepared to explain your reasoning.
 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.
Advances: Listening, Speaking
Supports accessibility for: Conceptual Processing, SocialEmotional 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

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:
 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 thirtytwos 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?)
Cooldown: Partial Quotients (5 minutes)
CoolDown
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.