Dividing Powers of 10

5 min

Teacher Prep
Setup
5 minutes of quiet work time. Brief whole-class discussion.

Narrative

In this activity, students investigate fractions that are equal to 1. This concept helps students make sense of the exponent division rule explored in a following activity. It is expected that students will try to compute the numerator and denominator of the fraction. Monitor for students who instead make use of structure to find factors in the numerator and denominator that can be used to show multiplication by 1 (MP7). 

Launch

Give students 5 minutes of quiet work time. Expect students to attempt to work out all of the multiplication without using exponent rules. Follow with a brief whole-class discussion.

Student Task

What is the value of the expression? Be prepared to explain your reasoning.

25⋅34⋅322⋅36⋅24\displaystyle \frac{2^5\boldcdot 3^4 \boldcdot 3^2}{2 \boldcdot 3^6 \boldcdot 2^4}

Sample Response

The expression is equal to 1. Sample reasoning:

  • The numerator and denominator both compute to 23,328 and 23,32823,328=1\frac{23,328}{23,328}=1. 
  • The numerator and denominator both have 5 factors that are 2 and 6 factors that are 3. Since the numerator and denominator have the same value, the entire fraction is equal to 1.
Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to see that a fraction is often easier to analyze when dividing matching factors from the numerator and denominator to show multiplication by 1. Invite students to share their answer and reasoning. If not brought up in students’ explanations, provide the following example and ask students how it could be used in this situation:

2⋅3⋅7⋅115⋅3⋅7⋅11=3⋅7⋅113⋅7⋅11⋅25=1⋅25=25.\displaystyle \frac{2 \boldcdot 3 \boldcdot 7 \boldcdot 11}{5 \boldcdot 3\boldcdot 7 \boldcdot 11} = \frac{3\boldcdot 7 \boldcdot 11}{3\boldcdot 7 \boldcdot 11} \boldcdot \frac{2}{5} = 1 \boldcdot \frac{2}{5} = \frac25.

If time allows, ask students "What has to be true about a fraction for it to equal 1?" (The numerator and denominator must be the same value and something other than 0.) Then invite students to create their own fraction that is equivalent to 1 and has several bases and several exponents.

MLR7 Compare and Connect. After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the different approaches have in common? How were they different?” and “Why did the different approaches lead to the same outcome?”
Advances: Representing, Conversing
Standards
Building On
  • 5.NF.5.b·Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
  • 5.NF.B.5.b·Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence <span class="math">\(a/b = (n \times a)/(n \times b)\)</span> to the effect of multiplying <span class="math">\(a/b\)</span> by <span class="math">\(1\)</span>.
Building Toward
  • 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

15 min