Divide Using Partial Quotients
Warm-up: Number Talk: Divide (10 minutes)
The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for dividing a three-digit number by a two-digit number. These understandings help students develop fluency and will be helpful later in this lesson when students use an algorithm that uses partial quotients to divide larger three-digit numbers by two-digit numbers.
- Display one expression.
- “Give me a signal when you have an answer and can explain how you got it.”
- 1 minute: quiet think time
- Record answers and strategy.
- Keep expressions and work displayed.
- Repeat with each expression.
Find the value of each expression mentally.
- \(110 \div 10\)
- \(121 \div 11\)
- \(132 \div 12\)
- \(154 \div 14\)
- “What stays the same with each problem? What changes?” (The quotient is always 11. There are always 11 groups of the divisor. The dividend and the divisor change.)
Activity 1: Compare Solutions (20 minutes)
The purpose of this activity is for students to explain the steps for using an algorithm that uses partial quotients to divide a three-digit number by a two-digit number. Because the size of the dividends is larger and the numbers are less friendly, the problems encourage students to reflect about which partial quotients will be most efficient for making the calculations. Monitor for students who begin with partial quotients that are multiples of 10. This strategy helps make the calculations simpler and students have seen and used multiples of 10 for these calculations throughout the last several lessons.
When students share and compare their methods with their partner and with other groups, they explain and improve their calculations (MP3).
- Groups of 2, then 4
- “You are going to use the partial quotients algorithm to solve some problems and compare the steps you used with other students.”
- “Now, decide who will solve each problem, solve your problem, and explain it to your partner.”
- 3–5 minutes: independent work time
- 1–3 minutes: partner discussion
- “Now, find another group of 2 and compare your solutions. How are they the same? How are they different?”
- 3–5 minutes: small group work time
- Monitor for students who use different multiples for the quotients.
Use an algorithm using partial quotients to find the value of one of the quotients. Be prepared to explain how you found the quotient.
- Explain to your partner how you found the value of the quotient.
- Pair up with another group and compare your work.
- Ask selected students who used different strategies to divide.
- “What questions do you have about algorithms that use partial quotients?”
- Display solutions from student solutions for the quotient \(589 \div 19\).
- “How are the strategies the same? How are they different?” (They all subtract 1 group of 19 at the end. Before that, they all subtract multiples of 10.)
- “How is the 30 in the solution C represented in the partial quotients in solutions A and B?” (Solution A uses 20 and 10 and solution B uses 10, 10, and 10 more. These add up to 30.)
Activity 2: Estimate and Solve (15 minutes)
The purpose of this activity is for students to practice using an algorithm that uses partial quotients to divide multi-digit numbers by two-digit divisors. Before finding the quotient, students estimate the value of the quotient which both helps students decide which partial quotients to use and helps them evaluate the reasonableness of their solution (MP8). For example, if students estimate that the value of \(529 \div 23\) is a little more than 20, that means that a good choice for first partial quotient is 20. Whenever possible, ask students to explain the steps they are taking.
To add movement to the activity, after students have solved each problem, they can partner with other students who used different partial quotients or got a different solution. This gives students the opportunity to explain their reasoning to one another and make adjustments to their work, as needed.
Advances: Representing, Conversing
Supports accessibility for: Social-Emotional Functioning
- Groups of 2
- 5 minutes: independent work time
- 5 minutes: partner work time
- Monitor for students who use different sets of partial quotients for the same problem.
Estimate the value of each quotient. Then, use an algorithm using partial quotients to find the value.
A reasonable estimate for \(612 \div 34\) is:
A reasonable estimate for \(529 \div 23\) is:
A reasonable estimate for \(1,\!044 \div 29\) is:
- Ask 2–3 students to share their work for the same problem that shows different partial quotients.
- “How can you make sure that the whole number quotient you got at the end is reasonable?” (It should be close to my estimate. I can multiply the quotient and divisor and that should give me the dividend.)
- If students pair and share with other partners, ask, “How did explaining your work to others help you today?” or “What did someone say today that helped you in your understanding of division?” (I learned that it’s ok to take more steps because I was comfortable with the multiples I used.)
“Today we used partial quotients to divide whole numbers.”
“What makes sense to you about this procedure?” (I get to remove multiples I’m comfortable with. The way the numbers are recorded makes it clear what is happening at each step.)
“What questions do you still have about using this procedure?” (What do I do if the numbers are bigger and more difficult? What if I try to subtract too much?)