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Unit 4, Lesson 4

Positive Rational Exponents

Algebra 2

4.1

Warm-up

10 mins

Math Talk: Regrouping Fractions

Standards Alignment

Building On

Addressing

Instructional Routines

Math Talk

MLR8: Discussion Supports

Materials

None

Activity 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

Activity

None

Student Task Statement

Find 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:

“First, I because .”
“I noticed , so I .”

Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

4.2

Activity

15 mins

You Can Use Any Fraction as an Exponent

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

4.3

Activity

10 mins

Fractional Powers Are Just Numbers

Instructional Routines

None

Materials

To Gather

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.)

to access Cool-down.

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

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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.

Activity

None

Student Task Statement

Use exponent rules to explain why these expressions are equal to each other:
Write using radicals.
Write using radicals. Show your reasoning using exponent rules.

Student Response

to access Student Response.

Building on Student Thinking

If students do not yet correctly write the expressions using radicals, consider saying:

“Tell me more about why the first three expressions are equivalent to each other.”
“Where do you see a cube root in ?”

Are You Ready for More?

Launch

Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.

Provide access to scientific calculators. Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.

Activity

None

Student Task Statement

Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)

	0	1	2
(using exponents)
(decimal approximation)

Plot the points that you filled in.

$<p>Horizontal axis, 0 to 2, by the fraction 1 over 3’s. Vertical axis, 0 to 4, by 0 point 2’s.</p>$
Connect the points as smoothly as you can.
Use this graph of to estimate the value of the other powers in the table, and write your estimates in the table.

Select one of the columns of the table that includes one of your estimates.
1. Write the power from the second row of the column you chose using radical notation.
2. What is the exact value of that number cubed?
3. Raise your decimal estimate from the table to the third power. What should it be? How close did you get?

Student Response

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Unit

Lesson

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Assessment Access

Lesson 4

Positive Rational Exponents

Warm-up

Math Talk: Regrouping Fractions

Standards Alignment

4.NF.B

HSN-RN.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

You Can Use Any Fraction as an Exponent

Instructional Routines

Materials

Activity

Fractional Powers Are Just Numbers

Instructional Routines

Materials

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

HSN-RN.A.2

Standards Alignment

HSN-RN.A.1

HSN-RN.A.2

Standards Alignment

HSF-IF.C.7.e

HSN-RN.A.1

HSN-RN.A.2