Describing Large and Small Numbers Using Powers of 10

5 min

Teacher Prep
Setup
Groups of 2. Give students 2 minutes of quiet work time followed by partner and whole-class discussions.

Narrative

In this Warm-up, students connect thousand, million, billion, and trillion to their respective powers of ten—103,106,10910^3, 10^6, 10^9, and 101210^{12}. Understanding powers of 10 associated with these denominations will help students reason about quantities in real-world contexts, such as the number of cells in a human body (trillions) or the world population (billions).

Launch

Arrange students in groups of 2. Give students 2 minutes of quiet work time and 1 minute to compare their responses with their partner. Given the limited time, it may not be possible for students to create examples for each of the values in the second question. Tell students to try to at least find 1 or 2 examples and then to find others as time allows. Follow with a whole-class discussion.

Student Task

  1. Match each expression with its corresponding value and word.
    expression
    10-310^{\text-3}
    10610^6
    10910^9
    10-210^{\text-2}
    101210^{12}
    10310^3

    value
    1,000,000,000,000
    1100\frac{1}{100}
    1,000
    1,000,000,000
    1,000,000
    11,000\frac{1}{1,000}

    word
    billion
    milli-
    million
    thousand
    centi-
    trillion

  2. For each of the numbers, think of something in the world that is described by that number.

Sample Response

  1. expression value word
    10-310^{\text-3} 11,000\frac{1}{1,000} milli-
    10610^6 1,000,000 million
    10910^9 1,000,000,000 billion
    10-210^{\text-2} 1100\frac{1}{100} centi-
    101210^{12} 1,000,000,000,000 trillion
    10310^{3} 1,000 thousand
  2. Answers vary.
Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to get a sense of the comparative sizes of very large and very small numbers by connecting them to a concrete example. Begin by displaying the completed table for all to see and address any questions or disagreements. Then invite students to share their examples for the final question. After each student shares, ask the class whether they agree that the given example could be described by that value.

If necessary, consider sharing some of the following examples:

  • Thousandth (10-310^{\text-3})
    • A milliliter is 11,000\frac{1}{1,000} of a liter.
  • Hundredth (10-210^{\text-2})
    • A centimeter is 1100\frac{1}{100} of a meter.
    • A penny is 1100\frac{1}{100} of a dollar.
    • A yard is 1100\frac{1}{100} of the distance between opposing end zones on an American football field.
  • Thousand (10310^3)
    • Population of an endangered species
    • Gallons of fuel it would take to fill 50 cars
  • Million (10610^6)
    • Population of the state of Delaware in 2022
    • Acres covered by the state of Rhode Island
    • Number of seconds in 12 days
  • Billion (10910^9)
    • Population of India or China (each ~1.4 billion)
    • Number of seconds in 31 years
  • Trillion (101210^{12})
    • Number of red blood cells in one-half pint of blood
    • Number of stars in a giant galaxy
    • Number of seconds in 31,688 years
Anticipated Misconceptions

If students confuse the prefix “milli-” with the word “million,” consider explaining that:

  • The word “million” literally means “a big thousand,” and so both “million” and “mille” are related to the Latin “mille,” meaning “thousand.”
  • While “milli-” is talking about a thousand parts (thousandths), “million” is talking about a thousand thousands.
Standards
Building On
  • 5.NBT.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • 5.NBT.A.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Building Toward
  • 8.EE.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <em>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger.</em>
  • 8.EE.4·Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
  • 8.EE.A.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <span>For example, estimate the population of the United States as <span class="math">\(3 \times 10^8\)</span> and the population of the world as <span class="math">\(7 \times 10^9\)</span>, and determine that the world population is more than <span class="math">\(20\)</span> times larger.</span>
  • 8.EE.A.4·Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

15 min

15 min