In this activity, students explore patterns to discover the rule that (MP8). Students will work only with cases where   and will extend the rule to include cases where or \(n in following activities and lessons. 

The goal of this discussion is to reinforce the exponent rule for dividing powers of 10. Introduce and explain the visual display prepared earlier. This display should be kept visible to students throughout the remainder of the unit. 

Continue to reinforce student understanding of this rule by writing out an expanded form of each expression when discussing the following questions:

• “What is written as a single power of 10?” ( )
• “When writing as a single power of 10, how many factors of 10 from the numerator and denominator can be re-written as a fraction equivalent to 1?” (20 factors of 10 from the numerator and 20 factors of 10 from the denominator can be written as a fraction equivalent to 1.)
• “What is as a single power of 10?” ( )

In this activity, students extend exponent rules to discover why it makes sense to define as 1 (MP8). Students also critique a statement that is intentionally incorrect and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

The purpose of this discussion is to solidify the concept that . Invite students to share their answers to the questions, and then ask them to share their thinking about what means.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:  “ means zero factors of 10 and so must equal 0.”
• Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

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

This activity expands on a previous Cool-down as students generate different representations of the same number to solidify what they have learned about exponents. 

Students are not expected to make an exponent of 8 using negative exponents, but do not discourage it if they do. Explain to these students that, while the rules still work when using negative exponents, it is not yet clear what the value of  is, and this will be explored in following lessons.  

The goal of this discussion is for students to share different expressions that have the same value as . Tell students to compare responses with their partner, and invite students to share examples that show creativity or that combine multiple rules together. 

To include more students in the discussion, consider asking: 

• “What is similar or different between yours and your partner’s representations?” 
• “How can you build on ’s method to come up with another expression?”