In this lesson, students build on what they know about positive fractional exponents, exponent rules, and graphs in order to make sense of negative fractional exponents. Like they did with positive fractional exponents, students graph \(y=2^x\) for negative integer values of \(x\), sketch the continuous curve between those points, and estimate the \(y\)coordinates on the curve for various negative rational \(x\)coordinates. This leads to the observation that expressions like \(b^{\textx}\) can be rewritten as \(\dfrac{1}{b^x}\) for any rational number \(x\).
Students make use of structure when they use graphs to estimate the value of exponential expressions and use exponent rules to test the accuracy of their estimates (MP7).
Lesson overview
 5.1 Warmup: Math Talk: Don’t Be Negative (5 minutes)
 5.2 Activity: Negative Fractional Powers Are Just Numbers (15 minutes)

5.3 Activity: Any Fraction Can Be an Exponent (15 minutes)
 Includes "Are you Ready for More?" extension problem
 5.4 Activity: Make These Exponents Less Complicated (10 minutes)
 Lesson Synthesis
 5.5 Cooldown: Switch It (5 minutes)
Learning goals:
 Draw a graph representing negative rational exponents and use it to estimate values.
 Use both radicals and exponents to represent numbers.
Learning goals (student facing):
 Let’s investigate negative exponents.
Learning targets (student facing):
 I can interpret exponents that are negative fractions.
Required Materials:
 Scientific calculators
Standards:
 This lesson builds on the standard:CCSS.8.EE.A.1MS.8.EE.1MO.8.EEI.A.1
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