The purpose of this activity is to develop students’ understanding of equivalent expressions. Students use tape diagrams to represent pairs of algebraic expressions when a one-digit whole number is given for the variable. Students see that expressions may be equal at some values for the variables, but not at others. Through repeated reasoning, they also begin to generalize that equivalent expressions are expressions that are equal at all values (MP8).

The goal of this discussion is to make sure students understand what it means for two expressions with a variable to be equivalent.

 Invite several students to share the two diagrams they drew for and . Ask:

• “How are the two diagrams in each pair alike?” (The two diagrams are always the same length regardless of the value of chosen.) 
• “How are the two diagrams in each pair different?” (The placement of and 3 are not the same.)

Display the equation . Point out that we can see that this equation is always true regardless of the value of . We call and equivalent expressions, because their values are equal no matter what the value of is. 

Display the equation , and ask students to consider this equation as they review the diagrams they drew on the grids for the first four problems. Point out that we can see that this equation is true when is 1, but not for the other values of that we tried. So, we can say that is equal to when is 1. Ask:

• “How do you know that and are not equivalent expressions?” (They are not equal for every value of .)

In this activity, students apply what they know about the meaning and properties of operations to deepen their understanding of “equivalent expressions.” The focus is on looking for and making use of structure (MP7), rather than identifying all the types of equivalent expressions appropriate to grade 6.

Monitor for students who apply their knowledge of operations with numbers to reason about operations with variable expressions. For instance:

• Having learned that  is equivalent to , students then reason that . 
• Knowing that , students reason that . 

Students may also use diagrams to reason or to explain their reasoning.

Invite students to share their pairs and reasoning, including those who used diagrams, if any. To highlight the idea that expressions with variables are equivalent only if they have the same value for all values of the variable, ask questions such as:

• “How can you be sure that the expressions on both sides will have the same value no matter what value is used for ?”
• “Are there two expressions that you knew right away are equivalent, without needing to test values or draw diagrams? If so, which ones, and how did you know?”