In this activity, students use what they know about operations to identify multiple equations with variables that match a given diagram. For example, the relationship between the quantities 2, 5, and 7 expressed by the equation can also be written as , , , and Ask students to explain how the parts of these equations match the parts of the tape diagram.

Note that the word “variable” is not defined until the next lesson. It is not necessary to use that term with students yet. Also, this activity includes the equivalent expressions and because these equivalent ways of writing it should be familiar from earlier grades, but the term equivalent expressions is defined more carefully later in this unit. Even though this familiar example appears, the general idea of equivalent expressions is not a focus of this lesson.

Throughout the activity, students have opportunities to use both informal and more formal language to describe equations (MP6).

Focus the discussion on why the same relationship can be expressed in more than one way. Elicit ideas such as:

• A multiplicative relationship can also be expressed using division.
• An additive relationship can also be expressed using subtraction. 
• It does not matter how expressions are arranged around an equal sign. For example, and mean the same thing.
• Repeated addition (or expressions with equal addends) can also be represented with multiplication. For example, is another way to express

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they matched each equation to a diagram. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “4 added to is 12” already on it and can be updated with the more precise phrase “4 added to the unknown value is equal to 12.”) In the next lesson, students will learn even more precise language to describe the relationships represented by the equations. They will revisit this display to update that language.

Consider asking questions, such as:

• “How can you tell if a diagram represents addition or multiplication?”
• “Once you were sure about one equation, how did you find others that matched the same diagram?”
• “What is the same about the equations that represent the same diagram? What is different?”

In this activity, students draw tape diagrams to match given equations. Then they reason about the unknown value and what it represents in each diagram and equation. Although some students may be tempted to identify the value of the variable, that should not be the focus of this activity. Students will focus on what it means to solve an equation in the next lesson.

• “What does mean?”
• “How could you represent 3 equal groups with unknown size y?”

Students might think they need to show an unknown number () of equal groups of 3. While this is possible, showing 3 equal groups with unknown size is simpler to represent. 

Invite students to share their diagrams and explanations of the meanings of and . Emphasize explanations that use the terms “unknown number,” “unknown value,” or “unknown quantity.” 

To highlight how a tape diagram represents the quantities and relationship shown in each equation, ask questions such as:

• “Where do you see the addition (or multiplication) in this diagram?”
• “Where do you see the 3 in the tape diagram representing the multiplication equation?”
• “How is the equal sign represented in the diagram?” (The parts have the same value as the whole or total. The numbers and letters in the boxes add up to the total shown for the whole rectangle.)