Representing Exponential Growth

10 min

Narrative

This Math Talk focuses on properties of exponents. It encourages students to think about how to multiply and divide numbers in exponential notation when the bases are the same and to rely on the structure of the notation to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students work with exponential growth.

Launch

Before starting the math talk, it may be helpful to take time to ensure that students understand the question. Ask students for examples and non-examples of a power of 2. Some examples are 252^5 and 21002^{100}. Non-examples include 1002100^2 and 5⋅25\boldcdot 2. It may be useful to further remind students that, for example, 252^5 equals 2⋅2⋅2⋅2⋅22 \boldcdot2 \boldcdot2 \boldcdot2 \boldcdot2.

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

Rewrite each expression as a power of 2.

  • 23⋅242^3 \boldcdot 2^4
  • 25⋅22^5 \boldcdot 2
  • 210÷272^{10} \div 2^7
  • 29÷22^9 \div 2

Sample Response

  • 272^7. Sample reasoning: (2⋅2⋅2⋅2)⋅(2⋅2⋅2⋅2)=2⋅2⋅2⋅2⋅2⋅2⋅2⋅2(2 \boldcdot 2 \boldcdot 2 \boldcdot 2) \boldcdot (2 \boldcdot 2 \boldcdot 2 \boldcdot 2) = 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2
  • 262^6. Sample reasoning: 25⋅2=(2⋅2⋅2⋅2⋅2)⋅22^5 \boldcdot 2 = (2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2) \boldcdot 2
  • 232^3. Sample reasoning: Ten 2s are multiplied in the numerator of the fraction and seven are in the denominator. That can be split into a fraction with seven 2s in both the numerator and denominator, with three 2s left over that are multiplied by the fraction. The fraction has a value of 1, which is multiplied by the left over 2s (which can be written as 232^3), so the result is 1⋅231 \boldcdot 2^3 (or simply 232^3).
  • 282^8. Sample reasoning: This can be written as 22⋅28=1⋅28\frac{2}{2} \boldcdot 2^8 = 1 \boldcdot 2^8
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?”

Remind students that exponents can be added when multiplying exponential expressions with the same base and subtracted when dividing those with the same base.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because. . . .” or “I noticed _____ 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
Building On
  • 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>

15 min

10 min