Lesson 2

The purpose of this warm-up is to re-introduce students to these diagrams as a representation of relationships between quantities. As students use tape diagrams as a tool for reasoning, they understand that the length of a piece of the “tape” carries meaning. Two pieces drawn to be the same length are understood to represent the same value. These pieces can be labeled with values to clarify what is known about the diagram, so two pieces labeled with the same letter indicate that they have the same value, even if that value is not known. These diagrams will be helpful for reasoning about situations in activities in this lesson. When students choose to use a tape diagram to represent a relationship between values and reason about a problem, they are using appropriate tools strategically (MP5). Tasks like this one ensure that students understand how such a tool works so that they are more likely to choose to use it correctly and appropriately.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Give students 1 additional minute of quiet work time to complete the second question followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class whether they agree or disagree and to explain alternative ways of thinking, referring back to the images each time.

Ask students to share possible values for the variables in each diagram. Record and display their responses for all to see. If possible, record the values on the displayed diagram. If the idea that pieces labeled with the same variable represent the same value does not arise in the discussion, make that idea explicit. For example, students should assume that all the pieces labeled with \(y\) in one diagram have the same value. When they make tape diagrams, they know to draw rectangles of the same length to show the same value, but since quick diagrams are sometimes sloppy, it’s also important to label pieces with numbers or letters to show known and relative values.

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 \(x\) and \(y\) 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.

Arrange students in groups of 3. (Some groups of 2 are okay, if needed.)

Ask students if they know what a “flyer” is. If any students do not know, explain or ask a student to explain. If possible, reference some examples of flyers hanging in school.

Ensure students understand they should take turns speaking and listening, and that there are two things to do for each diagram: explain why it represents the story, and also figure out any unknown values in the story.

Students may not realize that when a variable is assigned to represent a quantity in a situation, it has the same value each time it appears. Revisit what \(x\) and \(y\) represent in these problems and why each occurrence of a variable must represent the same value.

In the second situation, students might argue that a more accurate representation would be 5 boxes with \(y\) to show the first distribution of stickers, and then five boxes with 2 to show the second distribution. Tell students that such a representation would indeed correctly describe the actions in the situation, but that the work of the task is to understand this diagram to set us up for success later. 

Tape diagrams represent relationships between quantities in stories. The goals here are to make sure students understand how parts of the diagram match the information about the story, and for them to begin to reason about how the diagrams connect to the operations that can help 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, ask the class if anyone thought about it a different way. (One additional line of reasoning for each diagram is probably sufficient.)

Here are some questions you might ask 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 3 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 the previous activity, students interpreted given tape diagrams and explained how they represented a story. Here, they have a chance to draw tape diagrams to represent a story. The first story is a bit more scaffolded because it specifies what \(x\) represents. In 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.

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 \(x\) stickers, but the other stories do not define a variable. 
• Display one diagram for each story and ask students to explain how they are alike and how they are different.