Analyzing Graphs

5 min

Narrative

This Warm-up encourages students to look for patterns in real numbers, namely the decimal expansions of powers of 12\frac{1}{2}. Students are likely to recognize the 5, 25, 125 in the decimal expansion as powers of 5, and use that insight to describe the pattern, to extend it, and to notice other related patterns. In this Warm-up, they are not expected to reason about why the non-zero digits of the decimal expansions of powers of 12\frac12 are related to powers of 5.

Launch

Arrange students in groups of 2. Encourage them to think quietly for a minute before sharing their responses with their partner.

Student Task

fraction 12\frac12 14\frac14 18\frac18     116\frac{1}{16}         132\frac{1}{32}    
decimal 0.5 0.25 0.125

In the table, find as many patterns as you can. Use one or more patterns to help you complete the table. Be prepared to explain your reasoning.

Sample Response

fraction 12\frac12 14\frac14 18\frac18 116\frac{1}{16} 132\frac{1}{32}
decimal 0.5 0.25 0.125 0.0625 0.03125

Sample responses:

  • Each number is half of the previous number. We can find the next number by multiplying the current number by 12\frac12 (or by 0.50.5), or by dividing by 2.
  • The non-zero digits of the decimals are powers of 5 (51, 52, 53, etc.), and each time there is an additional decimal place.
  • 0.5 can be written as 510\frac{5}{10}, 0.25 as 25100\frac {25}{100}, and 0.125 as 1251,000\frac{125}{1,000}. The numerators are powers of 5 and the denominators are powers of 10, so the next two numbers are 54104\frac{5^4}{10^4}, which is 62510,000\frac {625}{10,000} or 0.0625, and55105\frac{5^5}{10^5}, which is3,125100,000\frac {3,125}{100,000} or 0.03125.
  • The denominators in the fractions are powers of 2.
Activity Synthesis (Teacher Notes)

Select a few students (or groups) to share their responses. If not already mentioned by students, discuss:

  • “Are the successive numbers exhibiting linear or exponential change? Explain your reasoning.” (Exponential because each number is being multiplied by 12\frac{1}{2} or 0.5 to get to the next one.)
  • “Are the successive numbers getting larger or smaller? Explain your reasoning.” (Smaller because 1 is being divided by greater and greater numbers.)

If time permits, consider asking: “Why do powers of 5 show up in the decimals?” (Because each number is being multiplied by 0.5 or 510\frac{5}{10}.)

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

15 min