In this activity, students explain how a tape diagram represents a situation. They also use the tape diagram to reason about the value of the unknown quantity. Students are not expected to write and solve equations here; any method they can explain for finding values for and is acceptable. While some students might come up with equations to describe the diagram and solve for the unknown, there is no need to focus on developing those ideas at this time.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

The goals of this discussion are to make sure students understand how parts of the diagram match the information given in the story, and to encourage students to start reasoning about how the diagrams connect to the operations they can use to find unknown amounts.

Invite one group to provide an explanation for each diagram—both how the diagram represents the story, and how they reasoned about the unknown amounts. After each situation is discussed, ask the class if anyone thought about it a different way. (One additional line of reasoning for each diagram is sufficient.)

Consider asking some of the following questions to encourage students to be more specific:

• “What question could you ask about the story?”
• “Where in the diagrams do you see equal parts? How do you know they are equal?”
• “What quantity does the variable represent in the story? How do you know?”
• “In the first story, where in the diagram do we see the ‘remaining flyers’?”
• “Why don’t we see the number 3 in the first diagram to show the three remaining volunteers?”
• “In the second diagram, where are the five volunteers represented?”
• “How did the diagrams help you find the value of the unknown quantities?”

In this activity, students draw tape diagrams to represent three stories. The first story is a bit more scaffolded because it specifies what x represents. For the other two stories, students need to decide which quantity to represent with a variable and choose a letter to use. As with all activities in this lesson, students are not expected to write and solve an equation. This preliminary work supports the understanding needed to be able to represent such situations with equations in later lessons. Since students are defining a quantity of interest and assigning a variable, they are engaging in aspects of mathematical modeling (MP4).

In the digital version of the activity, students use an applet to create their tape diagrams. The applet allows students to divide the tape into equal or unequal parts and label the parts and total. Demonstrate how the controls work in the applet, especially how the tape can be divided into 2 sections and the divider between those sections can be moved to adjust the sizes. Each section can then be divided into equal parts. The digital version may be helpful for students who struggle with drawing their own diagrams. However, even if the class is using the digital version, some students may prefer to draw the diagrams in their notebooks or on scratch paper.

Much of the discussion will take place in groups. Here are some ideas for synthesizing students’ learning about creating tape diagrams:

• Ask students if they had any disagreements in their groups and how they resolved them.
• Ask students how they decided which unknown quantity to find in the story. The first story specifies stickers, but the other stories do not define a variable. 
• Display one diagram for each story and ask students to explain how the diagrams are alike and how they are different.