Unit 6, Lesson 7
Negative Exponents
Extra Support Materials for Algebra 1
7.1
Warm-up
Standards Alignment
Building On
Addressing
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 dividing powers of ten and using exponents to rewrite powers of ten. It encourages students to think about exponents and rely on what they know about solving equations to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will divide expressions using exponents that have the same base.
In this activity, observations about subtracting exponents in the first three equations suggests a method for solving the last equation. Students may be puzzled by the last question. The purpose of including this equation is to conclude that the solution must be -1 because the left side can be rewritten as . The rest of this lesson offers more opportunities to make sense of negative exponents, so it’s not necessary for all students to feel completely comfortable with this idea, yet.
To solve these equations, 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
NoneSolve each equation mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for using properties of exponents to solve problems. 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?”
Draw particular attention to methods that express 100 and 1,000 as powers of 10. For example, if we rewrite as , we can remember our properties of exponents that tell us the denominator must be equal to . The same property of exponents can be used to deduce that the solution to the last equation is -1.
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
7.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of an exponent of 0 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. Complete the first two rows together as an example. 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.
Activity
None7.3
Activity
Materials
NoneActivity Narrative
In this partner activity, students take turns using exponent rules to analyze expressions and identify equivalent ones. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2. Display the task for all to see. Four lists of expressions are given. Depending on the time available, ask students to work through all four lists, assign only a few lists to work through, or allow students to select which lists they would like to work through.
Tell students that, for each original expression, there is at least one equivalent expression in the list. One partner selects the expression or expressions they think are equivalent and explains their reasoning. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next original expression, the students swap roles. If necessary, demonstrate this protocol before students start working.
Give students 10 minutes to work followed by a whole-class discussion. As students work, notice the different strategies used to analyze the expressions in each list. Ask students that use contrasting strategies to share during whole-class discussion.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to identify equivalent expressions using properties of exponents. Display words and phrases, such as “exponent,” “base,” “zero power,” “the same base is multiplied, so the exponents are added,” and “the same base is divided, so the exponents are subtracted.”