The goal of this discussion is to see how the factors change as different numbers of steps are skipped. Ask students to share their responses. If it doesn’t come up, demonstrate that each entry can be rewritten using an exponential expression that fits a certain pattern, and that this insight can help us understand and describe the pattern. For example, each entry in the last table can be written as , where is the step number.

Here are some questions for discussion:

• “How do you calculate the factor from one entry to the next?” (I think about what I would multiply the first entry by to get the second entry. Or, I divide the second entry by the first.)
• “How does this factor change when looking at two steps? Three steps?” (The factor ends up being two or three powers of the same base.)
• “Some of the questions in this activity ended up having the same answer. Why?” (When we divide powers of the same base, we end up subtracting the exponents. Since the exponents were the same distance apart, we ended up with the same answer.)

The purpose of this discussion is to clarify the reasoning needed to complete the equivalent expressions. Display the correct answers and invite students to check their work. Direct students’ attention to the reference created using Collect and Display. Ask students to share their reasoning to complete the expressions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Select one or more examples and ask a student to share how they reasoned to get a correct answer and to provide a justification for each step. For example, these steps might be listed:

, by replacing the 16 with an equal expression,  

, by using a property of exponents to rewrite as