If students struggle to find a way to express consecutive even integers using variables, consider asking:

• “How did you decide if Andre was right?”
• “How could using variables to represent Andre’s idea about consecutive even integers help you be sure?”

The goal of this discussion is that students understand what an identity is and that using variables is a powerful tool that helps us state when a relationship is always true.

Invite students to share how they used variables to show that Andre is incorrect. It is important for students to understand that testing many values is not enough to prove that a relationship is always true. In the case of Clare, it would take only one pair of consecutive integers to show that her relationship is not always true. By using and to stand for any two consecutive integers, we can say with confidence that Clare's relationship is always true. Similarly, using and , we can say with confidence that Andre's relationship is not always true.

Tell students that Andre’s idea is close to an actual identity about consecutive even integers. Ask, “What could Andre change about his statement that would make it true?” (Andre could change “4 times” to “2 times” and then he is correct that the difference between the squares of two consecutive even integers will always be 2 times the sum of the two integers.)

Conclude the discussion by telling students that Clare's relationship is an example of an identity. An identity is a type of equation in which the expression on the left has the same value for all possible inputs as the expression on the right, making them equivalent expressions. Ask students if they can think of any other identities they have learned. If not brought up by students, tell them that and are both examples of identities, and Clare's relationship is actually a special case of the former for when .

If students have trouble keeping track of terms that can be combined, consider asking:

• “Can you explain how you rewrote your expression.”
• “How could using a diagram help you organize your work?”

The goal of this discussion is for students to generalize the work from the activity to , where is any positive integer.

Select 2–3 groups to share something they noticed about the expressions. After students describe how the product can always be written as just 2 terms, tell them that this phenomenon is sometimes referred to as “telescoping,” after the type of telescopes that could be collapsed when not in use. This is a feature that can be useful when working with expressions that have many terms.

Ask students to think about how to work this relationship the other way around. For example, is equivalent to . So what must be equivalent to? After some quiet work time, invite students to share their expressions and record them for all to see. It is important for students to understand that is equivalent to , not . It may be helpful to use a diagram to organize the multiplication and see that it is the product of and that results in . Students will use their understanding of this identity in a future lesson when they derive the formula for the sum of a geometric sequence.