standard_description: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
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