Lesson 6

The purpose of this Choral Count is for students to practice counting by 2 and 4 and notice common multiples of the two numbers in the count. Students may notice that the count by 4 is every other number in the count by 2. The relationship between the two counts can also be described using multiplication and this is the goal of the synthesis.

• “Count by 2, starting at 0.”
• Record as students count.
• Stop counting and recording at 30.
• Count by 4, starting at 0.
• Record as students count.
• Stop counting and recording at 60.

• “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.

• “How can I get the number in the second list from the corresponding number in the first list?” (Double it by adding it to itself or multiply the number by 2.)
• “How can I get the number in the first list from the corresponding number in the second list?” (I can take half of it or divide the number by 2.)

The purpose of this activity is for students to visualize and make sense of the context of problems they will solve in the next activity. They will also consider representations that can be used to model the quantities and actions in the situation (MP4) and try creating them. Students will not have time to make a complete representation, so the focus is on making sense of the situation and persevering in finding effective representations (MP1).

• Groups of 3 or 4
• Display the image of the lockers.
• “In your small group, discuss what you notice and wonder about the picture.”
• 1 minute: quiet think time
• 1 minute: group discussion
• Share and record responses. If students are not familiar with lockers, explain that lockers are used to hold students’ coats, books, and backpacks in some schools.

• Read the problem aloud to students.
• “Take some time on your own to make sense of the problem. You may want to try to visualize it or reread the problem.”
• 2 minutes: quiet think time
• Give students access to two-color counters, index cards, coins, and other materials that may be helpful to them.
• “Based on what you know, create a representation of the problem.”
• 5 minutes: independent work time
• Monitor for and select 3 different representations that:
  • represent lockers in different ways
  • show instances of contact with lockers (or of opening and closing)
  • show correspondence between each student and the lockers touched

• While discussing each representation, ask students:
  • “Where are the lockers in this representation?” (Numbers or pictures used to indicate lockers)
  • “How can we tell if a locker is open or closed in this representation?” (Symbols or colors used to indicate open or closed)
  • “How does this representation show which student is touching the lockers?” (Students and their actions listed in a table or a T-chart)
• “In the next activity, you will answer questions about the game. You may use your representation or one shared by a classmate to help you think through the problems.”

The purpose of this activity is for students to solve problems about a locker game by using the ideas of factors and multiples and by observing patterns in the numbers. The situation offers many possible explorations, but the questions around which lockers are touched and by how many people are designed to elicit understandings about multiplication.

Discussion about which lockers will be open or closed or by whom will likely move students away from thinking about factors and multiples and should not be the focus. The final question challenges students who are ready for more to look for additional patterns and generalize their observations. It is not expected that all students complete it. Students who are ready may simulate opening and closing the lockers in different ways, such as:

• using actual lockers, if available
• using a printed set of squares labeled 1 through 20
• using two color counters and 10-frames, if available
• using the head and tail sides of coins

• Groups of 3–4
• “Let’s solve some problems about a game you read about earlier, where students take turns opening and closing lockers.”
• “Silently read and think about each question.”
• 1 minute: quiet think time
• “Work in your group to answer each question. Consider the representations you created or saw earlier to help you think about the problems.”

• 15–20 minutes: group work time
• Monitor for students who:
  • represent each contact with the lockers in a systematic way
  • represent the open and closed lockers in various ways
  • reason about factors and multiples and use the vocabulary in their explanations
• Encourage students who use a pattern to answer the final question to share their explanation. Invite them to consider a situation with more lockers such as 50 or 100.

• “Take a few minutes to share your answers with students in a different group.”
• Invite students to share how they kept track of the number of touches for each locker.
• See lesson synthesis.