Lesson 21

The purpose of this warm-up is to elicit the idea that many different questions could be asked about this situation, which will be useful when students solve problems in a later activity. While students may notice and wonder many things about this situation, the various questions that could be asked about the situation are the important discussion points.

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

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses. 

• “What does it mean that some apples are packed into boxes and some are not?” (Some apples are in groups and some apples are just loose in one big group.)
• “What questions could we ask about this situation?” (How many apples did the farmer pick? How many boxes had apples in them? How many apples were in each box?)

The purpose of this activity is for students to think about what they need to know to solve two-step word problems. Students choose numbers that make sense together to complete the problem from the warm-up. They articulate relationships between the quantities in the problem to justify their number choices. If students quickly find a combination of numbers that work, encourage them to see if there are other possibilities or to write a completed situation with the numbers they have chosen.

Students who do not choose a matching set of numbers quickly make sense of and persevere in solving the problem as they consider the relationship between the different quantities and the restrictions that puts on which numbers can describe the situation (MP1).

• Groups of 2 and 4
• Keep the situation from the warm-up displayed.
• “Suppose the boxes the farmer packs are all the same size.” 

• “A list of numbers is shown in the activity. Work with your partner to choose 4 numbers that would make sense together in this situation. If you find one combination of numbers that works, you can look for other combinations.”
• 8–10 minutes: partner work time
• Groups of 4
• “Share with another group of students how your number choices make sense.”
• 2–3 minutes: small-group discussion

• Display this partially completed row in the table, such as: 

  total number of apples	number of apples not in boxes	number of boxes	number of apples in each box
  200	152	8

• “If you were given this information, how would you find the number of apples in each box?” (I could subtract 152 from the total of 200 and divide the answer by 8.)
• “What equation can we write to represent the situation in this example? Let’s use a letter for the quantity that we don’t know.”
• 1 minute: quiet think time
• Record equations that students wrote, for instance:
  •  \((200 - 152) \div 8 = n\)
  • \(8 \times n + 152 = 200\)
  •  \(152 + 8 \times n = 200\)

The purpose of this activity is for students to represent a problem with an equation using a letter for the unknown quantity and solve the problem. Students should be encouraged to use whatever strategy or representation makes sense to them. 

The synthesis focuses on student thinking for the first problem. Students might represent the situation with:

• a tape diagram or an area diagram 
• an equation that uses multiplication
• an equation that uses division

If students struggle to get started on a problem, encourage them to create a drawing or diagram. Students may also represent the situation or solve the problem before writing an equation if that makes more sense to them. While this activity is focused on independent practice, students can discuss with a partner if needed.

• “Now we’re going to solve some problems about an event at the apple orchard.”
• Survey the class on their familiarity with events or activities at farms or orchards.
• If students are familiar, ask: “What are some things you might see or do at a farm or an orchard?”
• If students are unfamiliar, share some activities that might take place at an orchard. Consider showing some images of a market or an event at an orchard.
• 1–2 minutes: partner discussion
• Share responses. 

• 8–10 minutes: independent work time 
• Monitor for:
  • a variety of strategies students use to solve or represent each problem
  • lingering questions or common misconceptions and how students overcome them 
• Consider asking: 
  • “How could you represent that?”
  • “What information do you know that might help you?”

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?”

• Select students who used different strategies to share their responses and reasoning for each problem.