Practice an Algorithm Using Partial Quotients
Warm-up: Which One Doesn’t Belong: Different Ways (10 minutes)
The purpose of this warm-up is to examine a variety of ways to use partial products to find a single quotient. As students use partial products to find more complex quotients, they need to be strategic about which multiples of the dividend to subtract. Small multiples may be easier for finding the partial products but it takes more of them to give a sum equal to the dividend.
- Groups of 2
- Display the image.
- “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
- 1 minute: quiet think time
- 2–3 minutes: partner discussion
- Record responses.
Which one doesn't belong?
- “Which strategy do you prefer for finding the value of \(1,\!312 \div 82?\)” (I like method C because it begins by taking out a pretty big multiple of 82.)
Activity 1: Find the Mistake (20 minutes)
The purpose of this activity is for students to identify and correct common errors in using an algorithm that uses partial quotients. One of the errors involves subtraction and two of them involve multiplication. Students may choose to correct the errors and continue the work that is there or they may choose to find the quotient in different way that makes sense to them. When students determine where the errors are and explain their reasoning, they critique and construct viable arguments(MP3).
Advances: Conversing, Representing
- Groups of 2
- 1–2 minutes: quiet think time
- 3–5 minutes: partner work time
- Monitor for students who:
- are able to clearly identify and describe the errors.
- are able to use the algorithm to calculate the quotients correctly.
Advancing Student Thinking
If students do not identify the mistakes, prompt them to evaluate the division expressions and ask, "How does your solution compare with the one in the activity?"
- Invite students to share where they notice errors and describe the errors.
- “What can you do to make sure you are not making the same errors in your calculation?” (Before I calculate, I can estimate. While I work on the problem, I can double check when I subtract or multiply. After I calculate, I can multiply the quotient by the divisor and make sure I get the dividend.)
- Encourage students to consider what they can do before, during, and after they calculate.
Activity 2: Practice Problems (15 minutes)
The purpose of this activity is for students to divide three- and four-digit dividends by two-digit divisors. As the size of the dividend increases, students have an option to subtract multiples of 100 of the divisor. In order to calculate efficiently, this becomes essential for a quotient like \(8,\!715 \div 21\) as it will take a lot of partial quotients that are multiples of 10 to reach the full quotient. Using an algorithm that uses partial quotients for larger numbers also requires fluency with subtraction.
Supports accessibility for: Memory, Organization
- Groups of 2
- 8-10 minutes: independent work time
- “When you check in with your partner, consider the following questions.”
- “How do the calculations represent each partial quotient?”
- “What suggestions can you give your partner to improve their work for next time?”
- 2-3 minutes: partner discussion
- Ask a few students to share their work.
- “How was your work similar to or different from your partner’s?” (We both tried to subtract multiples of 10 or 100 and then when the remaining amount was small we subtracted smaller multiples. My partner used fewer steps.)
- "What did you notice in your partner’s work that shows they understand how to use an algorithm that uses partial quotients?” (My partner picked multiples that could mostly be calculated in their head and worked on subtracting the partial quotients.)
“Today, we practiced using an algorithm using partial quotients to divide multi-digit numbers.”
“Tell your partner how to use an algorithm using partial quotients to find the value of \(935 \div 85\).” (Look for the biggest multiple of 10 and 85 that I can subtract from 935. Find \(10 \times 85\) and subtract the product from 935 to see how much more I have left to divide. Keep doing this until I get zero as a difference. To check my answer, multiply the quotient by the divisor to make sure I get back to the original dividend.)