In this activity, students use repeated reasoning to discover the rule  (MP8). When students articulate their reasoning for what happens when neither the bases nor the exponents are the same, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

The goal of this discussion is to clarify what happens when multiplying two factors with different bases that cannot be rewritten to have the same base.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “What happens if neither the exponents nor the bases are the same?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “ makes sense, but what do you mean when you say. . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Here is an example of a second draft: "When multiplying expressions where the bases are the same, the exponents can be added together. If the bases are different but the exponents are the same, the bases can be regrouped and multiplied. If the bases and the exponents are different, and the expressions cannot be rewritten to have the same base, then the bases can not be grouped together evenly, and the expression cannot be expressed with a single exponent."

Introduce and explain the visual display prepared earlier. This display should be kept visible to students throughout the remainder of the unit.  
 

This activity gives students an opportunity to deepen their thinking by generating different equivalent expressions using the rules of exponents. The process of generating different expressions requires students to look for and make use of structure when considering the numerous ways numbers can be broken into factors and how to combine those factors and express the result using exponents (MP7).

The purpose of this discussion is for students to share what they learned about the rules of exponents by working in groups and playing the game. Here are some questions for discussion:

• “Which exponent rule(s) did your group use the most? Why?”
• “What is something you learned about exponents from your group?”
• “How did your group change your strategy for finding equivalent expressions from one round to the next?”
• “Did you notice any patterns?”