The purpose of this activity is for students to equally divide sandwiches in situations where the number of portions does not evenly divide the number of sandwiches. Students select the number of people sharing and the number of sandwiches from a small set of deliberately chosen numbers. Depending on their choice, each share may be less than a full sandwich or more than a full sandwich. This encourages students to make sense of the problem and persevere in solving it (MP1). The focus of the Activity Synthesis is on different ways students choose to make equal shares and how they know that the shares are equal. Many of the approaches for dividing the sandwiches equally are examined in detail in the next activity. In this activity, students explain how they know the shares are equal. Student representations may include coloring to show the equal shares.

Monitor for and select students with the following approaches to share in the Activity Synthesis:

• Choose numbers so each person gets one or two whole sandwiches and a fraction of a sandwich.
• Choose numbers so each person gets less than a full sandwich.

The approaches will later be displayed side by side to help students relate equal shares of objects to division and to fractions. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example for each approach, look at the Student Responses.

• Invite previously selected students to display their work side by side without sharing their thinking.
• “Take a minute to look at each of these diagrams.”
• Connect students’ approaches by asking:
  • “Does each person get more than a full sandwich or less? How do you know?” (Less because there are more people than sandwiches or more because there are more sandwiches than people.)
• Connect students’ approaches to the learning goal by asking:
  • “This situation is about sharing. We can represent equal sharing situations with division expressions. What division expressions can we write to represent each of the situations we represented?” (, , , , , )

In the previous activity students chose numbers to represent the sandwiches and the people sharing them and then represented those equal shares. The purpose of this activity is for students to explain how different representations can show equal shares when a division situation results in a quotient that is a fraction. Students have an opportunity to use their own language to describe equal shares in a division situation. Students may name the numerical amount of sandwich that each person gets, but that is not the focus of this activity. In future lessons, students will focus on numerical solutions for division situations that result in a quotient that is a fraction.

This activity uses MLR2 Collect and Display. Advances: Reading, Writing.

• “How does the diagram represent the situation?”
• ”How might you use two rectangles to represent the situation in a different way?”

• Refer to the display.
• “These are the words, phrases, and expressions that you used when you were describing how you knew each person got the same amount of sandwich.”
• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display:
• “How does this expression represent the situation that Han drew?” (There are 2 sandwiches being shared by 3 people.)