# Lesson 12

An Algorithm Using Partial Quotients

## Warm-up: Notice and Wonder: Incomplete Solution (10 minutes)

### Narrative

The purpose of this warm-up is for students to analyze the structure of an algorithm that uses partial quotients, which will be useful when students use this method to divide multi-digit numbers in a later activity. Students have seen algorithms that use partial quotients in grade 4. The new aspect to these calculations in grade 5 is that the dividend is now a 2-digit number. While students may notice and wonder many things about this image, the relationship between multiplication and division and the purpose of subtraction are the important discussion points.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “This is an algorithm used to divide whole numbers. The algorithm is not complete. What might you do next?” (Find $$5 \times 16$$ and subtract it from 128.)

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

### Narrative

The purpose of this activity is for students to interpret a partial quotients calculation with a two-digit divisor. Before interpreting the partial products calculation, students find the value of the quotient in a way that makes sense to them. This will help them understand the partial quotients calculation by familiarizing themselves with the numbers and likely some of the steps in the calculation. In explaining both their answers and strategies and Elena's, students need to be precise in their word choice and use of language (MP6) and they also have an opportunity to improve their argument and critique the reasoning of others (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Writing, Speaking, Listening.

Representation: Develop Language and Symbols. Provide students with access to Jada’s and Clare’s methods from the Notice and Wonder routine in the previous lesson so that students can reference their methods when solving the division problem.
Supports accessibility for: Memory, Organization

### Launch

• Groups of 2
• “Pause your work after you find the value of $$448 \div 16$$.”

### Activity

• 3–5 minutes: independent work time
• “Discuss how you found the value of $$448 \div 16$$ with your partner.”
• 1–2 minutes: partner discussion
• “Describe the steps Elena took to find the value of $$448 \div 16$$.”
• 3–5 minutes: independent work time

### Student Facing

1. Find the value of $$448 \div 16$$. Show your thinking. Organize it so it can be followed by others.

(Pause for teacher directions.)

2. This is Elena’s work. Describe the steps Elena took to find the value of $$448 \div 16$$.

### Student Response

For access, consult one of our IM Certified Partners.

If students do not find the correct value of $$448 \div 16$$, ask, “ How is your work similar to and different from Elena’s work?”

### Activity Synthesis

MLR1 Stronger and Clearer Each Time
• “Share your description of how Elena found the value of $$448 \div 16$$ with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• (Optional) If needed, display question starters and prompts for feedback.
• “Can you give an example to help show . . . ?”
• “Can you use the word _____ in your explanation?”
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• “How was Elena’s strategy the same as and different from your strategy?” (She used the same calculations, but organized her work differently.)
• “How does the method Elena used help her organize her work?” (She records the multiples of 16 in one place and subtracts them in another.)

## Activity 2: Complete the Solution (15 minutes)

### Narrative

The purpose of this activity is for students to deepen their understanding of an algorithm that uses partial quotients and practice using it. Students use an algorithm that uses partial quotients to find quotients with a three-digit dividend and a two-digit divisor. Different levels of scaffolding are provided as some of the calculations are partly completed. If students struggle to decide what multiple of the divisor to subtract, encourage them to pick a multiple they can calculate easily and that is less than or equal to what remains of the dividend.

• Groups of 2

### Activity

• 8–10 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
• multiply by 10 to find the value of $$364 \div 14$$.
• multiply by multiples of 10 to find the value of $$364 \div 14$$.

### Student Facing

Use Elena’s strategy to complete the following problems:

### Student Response

For access, consult one of our IM Certified Partners.

If students do not complete the steps correctly, refer to the partially completed problems and ask them to describe the steps that are shown.

### Activity Synthesis

• Ask students to share the steps in the algorithm for the first problem and complete the problem.
• Ask previously identified students to share their work for $$364 \div 14$$.
• If needed, clarify any of the steps for using the algorithm.

## Lesson Synthesis

### Lesson Synthesis

Display student work for the quotient $$364 \div 14$$ in the last activity or use the work in the student responses.

“What was the first multiple of 14 that you subtracted from 364? How did you choose that multiple?” (I started with 10 because I knew that’s 140 and that I could subtract 140 from 364. I started with 20 because I knew that was 280 which was less than 364 but not too far from 364.)

“Why are multiples of 10 good choices for an algorithm using partial quotients?” (I know how to find them in my head.)

## Cool-down: What’s Next? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.