Unit 4, Lesson 5
Negative Rational Exponents
Algebra 2
5.1
Warm-up
Standards Alignment
Building On
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, .
Materials
NoneActivity 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
NoneFind 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 or to evaluate the last expression. If the results of the pre-assessment showed that students need extra practice with exponent rules, highlight any student strategies that leverage these rules and carefully record student thinking for all to see.
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
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
In this activity, students approximate the value of for various negative rational exponents by interpolating values on the graph of . Students test those estimates using their understanding of the connection between rational exponents and roots.
In this activity, students are asked to use their understanding of roots and exponents to answer the problems, so technology is not an appropriate tool.
5.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
Students practice identifying expressions involving rational exponents with equivalent radical expressions. In this activity, students are asked to use their understanding of roots and exponents to answer the problems, so technology is not an appropriate tool.
Launch
Since this activity was designed to be completed without technology, ask students to put away any devices.
Activity
None5.4
Activity
Optional
Instructional Routines
NoneMaterials
NoneActivity Narrative
Students build on their work in the previous activity to practice their understanding of rational exponents by finding expressions that have the same value.
Launch
Arrange students in groups of 2. Give students quiet work time before asking them to compare with their partner how they grouped the expressions. Follow with a brief, whole-class discussion.
Activity
NoneLesson 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.
| -1 | 0 | 1 | |||||
| (using exponents) | |||||||
| (equivalent expressions) | or | 1 | or | 5 |