Expressed in Different Ways

5 min

Narrative

This Math Talk focuses on using exponent rules. It encourages students to think about the equivalence of values with exponents and to rely on rules of exponents to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students adjust growth rates to different units of time.

To compare the values with exponents, students need to look for and make use of structure (MP7).

In describing their strategies, students need to be precise in their word choice and use of language (MP6). 

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 if each expression is equal to (1.21)100(1.21)^{100}.

  • ((1.21)10)10\left((1.21)^{10}\right)^{10}
  • ((1.21)50)50\left((1.21)^{50}\right)^{50}
  • ((1.1)2)100\left((1.1)^2\right)^{100}
  • (1.1)200(1.1)^{200}

Sample Response

  • It is equal. Sample reasoning: Using the rule of exponents with (ab)c=ab⋅c(a^b)^c = a^{b \boldcdot c}, this expression is equal to the given one.
  • It is not equal. Sample reasoning: This is equivalent to (1.21)50⋅50=1.212500(1.21)^{50\boldcdot 50} = 1.21^{2500}.
  • It is equal. Sample reasoning: 1.12=1.211.1^2 = 1.21, so this is equal to the given expression.
  • It is equal. Sample reasoning: (1.1)200=(1.12)100(1.1)^{200} = (1.1^2)^{100}, so it is equal to the previous expression which was also equal to the original expression.

Synthesis

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?”

Pay particular attention to the last expression, (1.1)200(1.1)^{200}. In order to identify this as equal to (1.21)100(1.21)^{100} students need to work backward and write this as ((1.1)2)100\left((1.1)^2\right)^{100}. Another approach would be to work forward and rewrite (1.1)200(1.1)^{200}  as 1.12⋅1001.1^{2 \boldcdot 100} The third expression is intended to facilitate this thinking. All of these problems rely on an important property of exponents, (xa)b=xab\left(x^a\right)^b = x^{ab}.

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, with a partner what they will say 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>
Building Toward
  • F-IF.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • F-IF.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • F-IF.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • F-IF.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • F-IF.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • HSF-IF.C.8·Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

15 min

15 min