standard_description: 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. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Activity
The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of zero exponents is also revisited. When students use tables to explore expressions with negative exponents, they are noticing and making use of structure (MP7).
Launch
Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Follow with a whole-class discussion.
Student Facing
Complete the table.
|                               | exponential form |   number form    |               calculations               |
|-------------------------------|------------------|------------------|------------------------------------------|
|                               |     \(2^5\)      |                  |                                          |
|                               |                  |        16        |                                          |
| \(\frac{2^4}{2}=2^{4-1}=2^3\) |     \(2^3\)      |                  |                                          |
| \(\frac{2^3}{2}=2^{3-1}=2^2\) |     \(2^2\)      |        4         |                                          |
|                               |                  |        2         |        \(4 \boldcdot \frac12=2\)         |
|                               |                  |        1         |        \(2  \boldcdot \frac12=1\)        |
|                               | \(2^{\text-1}\)  | \(\frac{1}{2}\)  |                                          |
|                               |                  | \(\frac{1}{4}\)  | \(\frac12  \boldcdot \frac12 = \frac14\) |
|                               | \(2^{\text-3}\)  |                  |                                          |
|                               | \(2^{\text-4}\)  |                  |                                          |
|                               |                  | \(\frac{1}{32}\) |                                          |
Student Response
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Activity Synthesis
Display the completed table, and resolve any questions students have. Questions for discussion:
“How would you make sense of
\(2^{\text-6}\)
?” (I know
\(2^{\text-5}=\frac{1}{32}\)
, and I know that
\(2^{\text-6}\)
would be the result of multiplying by another
\(\frac12\)
. So,
\(2^{\text-6} = \frac{1}{64}\)
.
“How could we figure out the value of
\(7^{\text-2}\)
?” (Possible response: We could rewrite it as
\(\frac{7^0}{7^2}\)
, and then
\(\frac{1}{49}\)
.


aligns: no