Lesson 18

The purpose of this What Do You Know About _____ is to invite students to share what they know and how they can represent division.

• Display the word division.
• “What do you know about division?”
• 1 minute: quiet think time

• Record responses.
• “How could we represent division?” (a drawing, with connecting cubes, an equation, an expression)

• “We have already learned a lot about division. We are going to continue to learn about division by working with larger numbers than we have before.”
• Consider asking: “What connections do you see between different answers?”

The purpose of this 5 Practices activity is to elicit students’ existing strategies for finding the value of quotients with larger numbers. Students should be encouraged to use whatever strategy or representation makes sense to them. 

Monitor and select students with the following strategies to share in the synthesis:

• make groups of 4 and see how many groups there are (with objects, drawings, arrays, or base-ten blocks)
• multiply up starting with \(4 \times10\)
• break the dividend up into tens and ones and divide each part

If appropriate, discuss connections between the strategies as they are shared, rather than after all strategies are shared. It is not essential that all the listed strategies are discussed, as students will consider these ideas in upcoming lessons. The main goal here is to elicit what students currently understand.

When students make sense of the contextual division problem they reason abstractly and quantitatively (MP2). Students who use the relationship between multiplication and division make use of structure (MP7).

• Display the statement: “On a field trip to the aquarium, a class of students is splitting into groups.”
• “What do you notice? What do you wonder?” (Students may notice: They are going to the aquarium in groups. They can’t go in one big group. Students may wonder: How many kids are in the class? How many kids are in each group? Are the groups the same size?)
• 1 minute: quiet think time
• Share and record responses.
• Give students access to connecting cubes or counters, base-ten blocks, and grid paper.

• “Solve this problem. Use whatever strategy or representation that works best for you.”
• 5 minutes: independent work time
• As students work, consider asking:
  • “Where are the groups of 4 in your work?”
  • “Where is the number of groups in your work?”
• “Share your response and strategy with your partner. Talk about how they are alike and how they are different.”
• 3 minutes: partner discussion

If students say they aren't sure how to start the problem, consider asking: “What is the problem about?” and “How could you represent the problem?”

• Select previously identified students to share in the order listed in the activity narrative.
• As each strategy is presented, invite the class to ask questions.
• Keep all the strategies displayed.
• “How are the strategies alike?” (All the strategies used \(4\times 10 = 40\) to help break the problem into smaller parts. The strategies with division and multiplication equations both had \(10 + 2 = 12\).)
• “How are the strategies different?” (In some strategies, students drew a representation. In others, students wrote multiplication or division expressions or equations.)

The purpose of this activity is for students to consider their strategies as they solve two other division problems involving equal groups with larger numbers. The divisor in the first problem is a low one-digit number. Students can see from the given situation that it is the number of groups. In the second problem, the divisor is a teen number, and the context suggests that it is the size of one group. Students are likely to adjust their strategy based on these observations. Focus the discussion on how students may have reasoned differently given a larger divisor or given what they understand about the situation.

• Groups of 2
• “Read the two problems about another field trip. Think about what strategies you might use to solve them.”
• 1 minute: quiet think time
• “Share your thinking with your partner.”
• 1 minute: partner discussion
• Give students access to connecting cubes or counters, base-ten blocks, and grid paper.

• “Work independently on the problems for a few minutes. Then, share your responses with your partner.”
• 5 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for students who use different representations (drew different diagrams or wrote different types of expressions or equations) for the two problems.

If students say they aren’t sure how to get started on the problem, consider asking: “What is the problem about?” and “How could you represent the problem?”

• Invite students to share their responses. Display or record their reasoning.
• Poll the class on whether they used a different strategy for solving the second problem than used for the first.
• Ask those who used a different strategy: “Why did you change your strategy?” (Sample responses:
  • In the first, the 3 represents 3 groups. In the second, the 12 is how many in each group.
  • In the first, the number used to divide is smaller. In the second, the number is larger.)