In this activity, students use expressions with variables to represent lengths of sides and areas of rectangles. These expressions are used to help students understand the distributive property and its use in creating equivalent expressions.

Solicit students’ responses to the first and third questions. Display two of the expressions, as shown. (Expressions that are equivalent to these are fine.) Ensure that everyone agrees that one expression is an acceptable response to the first question and the other is an acceptable response to the third question.

Ask students to look at the two expressions and invite them to share something they notice and something they wonder. Here are some things that students might notice.

• The 5 appears twice in one expression and only once in the other.
• These look like an example of the distributive property, but with a variable.
• These expressions must be equivalent to each other, because they each represent the area of the same rectangle.

If no students mention the last point—that the expressions are equivalent—ask them to discuss this idea.

In this activity, students are presented with several partitioned rectangles. They identify the length and width for each rectangle, and then write expressions for the area in two different ways, as: 

• The product of the length and the width, one of which is expressed as a sum of partial lengths. 
• The sum of the areas of the smaller rectangles that make up the large rectangle. 

Students reason that these two expressions must be equal since they both represent the total area of the partitioned rectangle (MP2). In this way, students see several examples of the distributive property. Students may choose to assign values to the variable in each rectangle to check that their expressions for area are equal. 

Select students to share their expressions for the areas of Rectangles D and F. Ask them to explain how each expression relates to the diagram. Display and annotate the diagrams, if possible. Ask students:

• “How do you know if the pair of expressions for the area of each rectangle are equivalent?” (They represent the area of the same figure. Applying the distributive property in one expression gives the other expression, so we know they are equivalent.) 
• “Suppose we know the value of the variable . What can you predict about the values of the two expressions?” (They would be equal.) “How would you test your prediction?” (Substitute the value for the variable and do the computation.)

Display the expressions from the rows for Rectangles D and F:

Tell students that as they work with a greater variety of expressions, it is helpful to be able to refer to the parts in the expression (just as students learned to use “factors” and “product” to refer to the parts in a multiplication, and “dividend,” “divisor,” and “quotient” for division.)

Introduce the word “term” to students. Explain that a term is a part of an expression. A term can be a single number, a single variable, or a product of numbers and variables. In the displayed expressions, 5, , , and are terms. Invite students to identify a few other terms in their completed table. 

Add “term” to the display of other vocabulary words from the unit (or revise similar words or phrases students used previously to name the same concept).