Unit 4, Lesson 5
Negative Rational Exponents
Algebra 2
5.1
Warm-up
5 mins
Math Talk: Don’t Be Negative
Standards Alignment
Building On
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example,
to see where instructional routines fit in your journey.
Materials
None
Activity Narrative
This Math Talk focuses on rational exponents. It encourages students to think about fractional and negative exponents and to rely on the properties of exponents to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students estimate values from an exponential graph.
To make sense of a negative fractional exponent, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneStudent Task Statement
Find the value of each expression mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for understanding fractional and negative exponents.
To involve more students in the conversation, consider asking:
- “Who can restate
’s reasoning in a different way?” - “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to
’s strategy?” - “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
If this strategy is not mentioned, consider asking students how they would use
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I
because .” - “I noticed
, so I .”
Advances: Speaking, Representing
5.2
Activity
15 mins
Negative Fractional Powers Are Just Numbers
Instructional Routines
to see where instructional routines fit in your journey.
5.3
Activity
15 mins
Any Fraction Can Be an Exponent
Instructional Routines
to see where instructional routines fit in your journey.
5.4
Activity
Optional
10 mins
Make These Exponents Less Complicated
Instructional Routines
to see where instructional routines fit in your journey.
Lesson Synthesis
In this lesson, students have worked with expressions that involve negative rational exponents. Here are some questions for discussion:
- “How are
and alike? How are they different?” (They are alike because they both involve an exponent of , which means the expression can be described using square roots. They are different because the negative exponent means that they are reciprocals of each other. In particular, while .) - “How could you write
using radicals?” - “What would be the result of cubing
?” (Cubing would give , which is .)
Student Lesson Summary
When we have a number with a negative exponent, it means we need to find the reciprocal of the number with the exponent that has the same magnitude, but is positive. Here are two examples:
The table shows a few more examples of exponents that are fractions and their radical equivalents.
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