Unit 4, Lesson 4
Positive Rational Exponents
Algebra 2
4.1
Warm-up
Standards Alignment
Building On
4.NF.B
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Addressing
HSN-RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define to be the cube root of because we want to hold, so must equal .
Materials
NoneActivity Narrative
This Math Talk focuses on multiplying fractions. It encourages students to think about multiplying fractions and whole numbers and to rely on the structure of the fraction to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students decompose fractions found in an exponent.
To multiply the values mentally, students need to look for and make use of structure (MP7) of commutativity and decomposition of rational numbers.
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 decomposing fractions such as .
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 no students mention the strategy of decomposing the fractions, bring it to their attention.
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
4.2
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
In this activity, students break down fractional exponents into a unit fraction times a whole number and rewrite the expressions using radicals. The first problem suggests how to break down fractions into a whole number times a unit fraction, and students can apply the same reasoning in subsequent questions.
Look for groups that write equivalent radical expressions that are different from each other to compare during the discussion.
When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Display the expression . Tell students that mathematicians have a convention for what this expression means, and invite them to make a conjecture about what it means. Emphasize that there are no wrong answers; we are just curious about what the expression might mean. Write, near the expression, each conjecture that is shared. Examples of conjectures might include , , or . Tell students that in this activity, they will use the properties of exponents to understand how mathematicians have decided to interpret this expression.
Arrange students in groups of 2. Tell students there are many possible answers for the questions.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the first question. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.
If time allows, display these prompts for feedback:
- “ makes sense, but what do you mean when you say ?”
- “Can you describe that another way?”
- “How do you know ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, follow with a whole-class discussion.
4.3
Activity
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph for a few integer values of , approximate a smooth curve passing through those points, and use that curve to estimate the value of for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
In the first problem of this activity, students are making sense of values with fractional exponents as numbers by plotting more familiar numbers, so technology is not an appropriate tool.
Lesson Synthesis
In this lesson, students extended the connection between roots and unit fraction exponents to describe other positive rational exponents in terms of roots. Building on this foundation, students can also generalize to fourth roots, fifth roots, and so on. Here are some questions for discussion:
- “Find the value of by rewriting the expression with an exponent.” ()
- “What are some different ways we can express 16 using exponents or roots? Think about how you might use fractions as exponents.” (, , , , , , and many more.)
Student Lesson Summary
Using exponent rules, we know is the same as because . But what about ?
Using exponent rules,
which means that
, so we could also write .
Here are more examples of exponents that are fractions and their equivalents:
| 0 | 1 | 2 | |||||
| (using exponents) | |||||||
| (equivalent expression) | 1 | or | 5 | or | or | 25 |