# Lesson 20

Strategies for Dividing

## Warm-up: Number Talk: Multiplication and Division (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for using multiplication to help them divide. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find the value of quotients.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$3 \times 5$$
• $$6 \times 5$$
• $$10 \times 5$$
• $$65 \div 5$$

### Activity Synthesis

• “How does thinking about multiplication help you divide?” (I can think about what number multiplied by 5 will be 65. I can break that into smaller products I know.)
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Ways to Divide (15 minutes)

### Narrative

The purpose of this activity is for students to transition from reasoning about division concretely or visually (using base-ten diagrams) to doing so more abstractly (by writing equations). It also reinforces the connections between multiplication and division.

Students make sense of three different strategies of dividing 78 by 3 and attend to the connections between the visual and numerical representations of the same quotient. As they do so, they practice reasoning quantitatively and abstractly (MP2).

During the synthesis, discuss how place value units play a role in all three strategies.

MLR8 Discussion Supports. Synthesis: Display Lin, Priya, and Tyler’s strategies. As students share their observations, annotate the display to illustrate connections. For example, annotate where students see the divisor, dividend, and quotient on each diagram.

### Launch

• Groups of 2
• “Take a couple minutes to make sense of Lin, Priya, and Tyler’s work.”
• 2 minutes: quiet think time

### Activity

• “Work with your partner to make sense of Lin, Priya, and Tyler’s work and to complete the activity.”
• 7–8 minutes: partner work time
• Monitor for students who:
• make connections between the equations and the base-ten diagram
• recognize how Tyler’s equations and reasoning are different from Priya’s equations
• Identify students who can explain these connections or distinctions to share during the synthesis.

### Student Facing

1. Lin, Priya, and Tyler found the value of $$78 \div 3$$. Their work is shown. Make sense of each student’s work.

Lin

Priya

\begin{align} 3\times 10&= 30\\ 3\times 10&= 30\\ 3\times \phantom{0}6 &= 18 \\ \overline {\hspace{5mm}3 \times 26} &\overline {\hspace{1mm}=78 \phantom{000}} \end{align}

Tyler

\begin{align} 3\times 20&= 60\\ 3\times \phantom{0}6 &= 18 \\ \\20 + 6 &= 26 \end{align}

2. How are the three students’ work alike?
3. How are they different?

### Activity Synthesis

• Invite students to share how Lin, Priya, and Tyler’s work are alike and how they are different.
• Select previously identified students to share additional connections that they noticed.
• “Why does it make sense that Priya and Tyler wrote multiplication equations to find the value of a quotient?” (Multiplication and division can represent the same equal-groups situation. To divide 78 by 3 is to find how many are in each of 3 groups or how many groups of 3 there are. We can multiply up to 78 to find the answer.)
• Consider asking: “What new ideas about dividing numbers did you learn and would like to try? Talk to a new partner about why you’d like to try them.”

## Activity 2: How Would You Divide? (15 minutes)

### Narrative

The purpose of this activity is for students to practice finding the value of division expressions using any strategy that makes sense to them. They may divide the dividend into equal groups or use the divisor to multiply up to the given dividend. They may choose to represent the division or multiplication with base-ten blocks or by drawing diagrams.

During the synthesis, highlight strategies that rely on place value, properties of operations, and the relationship between multiplication and division (MP7).

Representation: Develop Language and Symbols. Synthesis: Make connections between representations visible. Elicit from students the connections between the different strategies and representations shared by students.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing

### Required Materials

Materials to Gather

Materials to Copy

• Centimeter Grid Paper - Standard

• Groups of 2

### Activity

• “Find the value of each quotient. You can use whatever strategy or representation you prefer.”
• 8–10 minutes: independent work time
• Monitor for a variety of strategies and representations. Identify students to share in the synthesis.
• 2–3 minutes: partner discussion

### Student Facing

Find the value of each quotient. Explain or show your reasoning. Organize it so it can be followed by others.

1. $$80 \div 5$$
2. $$68 \div 4$$
3. $$91 \div 7$$

If you have time: Eighty-four students on a field trip are put into groups. Each group has 14 students. How many groups are there?

### Activity Synthesis

• Have previously selected students share their responses. Display or record the strategies and representations students use.
• For each problem, consider polling the class on the strategy they used.

## Activity 3: Compare, Divide within 100 [OPTIONAL] (10 minutes)

### Narrative

The purpose of this optional activity is for students to practice evaluating division expressions in order to make comparisons. Compare is a center that focuses on the procedural skills needed to solve single- and multi-step word problems. In this stage, students will use division to evaluate and compare quotients within 100.

This stage of the Compare center is used in grades 3, 4, and 5. When used in grade 3, remove the cards with two-digit divisors.

### Required Materials

Materials to Copy

• Compare Stage 4 Division Cards

### Required Preparation

• Create a set of cards from the blackline master for each group of 2. Remove the cards with two-digit divisors.

### Launch

• Groups of 2
• Give each group of students a set of cards from the blackline master.
• “Now let’s play Compare to practice what you learned about dividing numbers.”
• 1 minute: quiet think time
• Answer any questions about the directions. If needed, play a turn with the class.

### Activity

• 5–7 minutes: partner work time
• If time permits, provide extra time to play Compare.

### Student Facing

Play Compare with 2 players.

1. Shuffle the cards and split the deck between the players.
2. Each player turns over a card.
3. Compare the values. The player with the greater value keeps both cards.
4. Play until you run out of cards. The player with the most cards at the end of the game wins.

### Activity Synthesis

• Display: $$92 \div 4$$ and $$72 \div 3$$.
• “Suppose these are the two cards you draw. How would you decide which expression has the greatest value?”

## Lesson Synthesis

### Lesson Synthesis

“Today, we used a variety of strategies and representations to divide larger numbers. How do you like to divide larger numbers? Why?” (I like to use multiplication because I can use the multiplication facts I know to divide. I like to divide in parts because I can think about smaller division facts I know. I like to use drawings of base-ten blocks and think about putting them into equal groups because I can use tens and ones to divide.)