The goal of this discussion is to clarify how to rewrite expressions with roots and fractional exponents. Ask students to share their answers. Here are some questions for discussion:

• “Is it more challenging to begin with an expression with fractional exponents or an expression with roots?”
• “How can you use the exponent rules you know to show that ?” (, so . The exponent rule means that , which is the same as .)

The purpose of this activity is for students to apply their knowledge of fractional exponents to interpret an exponential situation. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). Students can use technology to reason about outputs for various inputs in an exponential equation in this activity before making sense of an exponential expression in the associated Algebra 1 lesson.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

The goal of this discussion is for students to connect their knowledge of exponential equations with their understanding of fractional exponents.

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

• Display this first draft:
  There were 13.1 fish 6 months after the scientists started monitoring.
  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.
• Display and review these criteria:
  • Use correct values for the number of fish and amount of time.
  • Use a whole number to represent the number of fish.
  • Measure in months or fractions of a year.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Ask students to share their solutions and reasoning. Consider highlighting different approaches including substituting values and graphing. Here are some questions for discussion:

• “If a scientist wanted to know the population of fish 3 months after they started monitoring, what -value could they use?” (, , or equivalent)
• “What does an -value of -1 mean in this situation?” (one year before the scientists started monitoring the fish population)