Practice with Rational Bases

5 min

Teacher Prep
Setup
Students in groups of 2–4. Display expressions for all to see. 1 minute of quiet think time, followed by small-group and whole-class discussions.

Narrative

This Warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the expression for all to see. Ask students to indicate when they have noticed one expression that doesn’t belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular expression doesn’t belong and together find at least one reason each expression doesn't belong.

Student Task

Which three go together? Why do they go together?

A

2825\dfrac{2^{8}}{2^5}

B

(4-5)8\left(4^{\text-5}\right)^{8}

C

(34)-5(34)8\left( \frac34 \right)^{\text-5} \boldcdot \left( \frac34 \right)^{8}

D

10-855\dfrac{10^{\text-8}}{5^5}

Sample Response

Sample responses: 

A, B, and C go together because:

  • They do not have 2 different bases.

A, B, and D go together because:

  • They all have bases with whole numbers.
  • They are not multiplying two factors together.

A, C, and D go together because:

  • They all have 2 bases.
  • They are not raising a power to another power.

B, C, and D go together because:

  • They all have an exponent that is negative.
Activity Synthesis (Teacher Notes)

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as "base," "factor," "power," and "rule," and to clarify their reasoning as needed. Consider asking the following questions:

  • “How do you know . . . ?”
  • “What do you mean by . . . ?”
  • “Can you say that in another way?”
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>

20 min

10 min