What about Other Bases?

5 min

Teacher Prep
Setup
Up to 1 minute of quiet work time per problem. Whole-class discussion.

Narrative

This Math Talk focuses on comparing powers of positive and negative numbers. It encourages students to think about repeated multiplication and to rely on what they know about integers and the meaning of the bases and exponents to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students find equivalent expressions involving exponents.

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies, and record and display their responses for all to see.
  • Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem. 

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Decide mentally whether each statement is true.

  • 35<463^5 < 4^6
  • (-3)2< 32\left(\text- 3\right)^2 < 3^2
  • (-3)3= 33\left(\text- 3\right)^3 = 3^3
  • (-5)2>-52\left( \text- 5 \right) ^2 > \text- 5^2

Sample Response

  • True. Sample reasoning: 353^5 has both a smaller base and a smaller exponent than 464^6, so its value will also be smaller.

  • Not true. Sample reasoning: Since a negative number multiplied by itself results in a positive number, both values are equal to 9.

  • Not true. Sample reasoning: A negative number multiplied by itself 3 times will result in a negative number, making (-3)3(\text- 3)^3 less than 333^3.

  • True. Sample reasoning: The expression on the left is equivalent to 25. The expression on the right could be written as -(5)2\text- (5)^2, meaning that only the 5 is squared and the resulting value is -25.

Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Addressing
  • 8.EE.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. <em>For example, 3² × 3<sup>-5</sup> = 3<sup>-3</sup> = 1/3³ = 1/27.</em>
  • 8.EE.A.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. <span>For example, <span class="math">\(3^2\times3^{-5} = 3^{-3} = 1/3^3 = 1/27\)</span>.</span>

10 min

15 min