### Summary

In this lesson, students will recall solving quadratic equations and rewriting radical expressions as expressions with rational exponents and vice versa. Students will learn how to solve equations with rational exponents and radicals. Then, students will apply this new knowledge to both mathematical and real-world problems. This is the second lesson of three in the "Radical Yet Rational" lesson series.

### Essential Question(s)

How do we use rational exponents and radicals to solve problems?

### Snapshot

**Engage**

Students recall rewriting and simplifying expressions with rational exponents and radicals.

**Explore**

Students solve quadratic equations and apply what they know to equations with rational exponents.

**Explain**

Students complete guided notes with the class and formalize their understanding of how to solve equations with rational exponents using either radicals or rational exponents.

**Extend**

Students apply what they have learned to solve real-world problems.

**Evaluate**

Students select three equations from a Choice Board and demonstrate their understanding of how to solve equations with rational exponents using either radicals or rational exponents.

### Instructional Formats

The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.

### Materials

Lesson Slides (attached)

Search and Solve handout (attached; one per pair; printed front only)

Guided Notes handout (attached; one per student; printed front only)

Guided Notes (Teacher Guide and Model Notes) (attached; for teacher use)

Real-World Rational Exponents handout (attached; one per pair; printed front only)

Choice Board handout (attached; one per student; printed front only)

Pencils

### Engage

5 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. Display **slide 3** to share the lesson’s essential question with students. Go to **slide 4** to share the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Go to **slide 5** and ask students to find a partner or assign student pairs. Pass out the attached **Search and Solve** handout to each pair of students.

Direct students’ attention to the first part of the handout: Find the Mistake. (Students will complete the second part of the handout during the Explore portion of this lesson.) Have students work in pairs to look through the given problem and find the mistake in the simplifying process. On the handout, ask students to describe and correct the error in their own words.

After giving students time to complete the task, ask for volunteers to share their thoughts. Use student responses to see which misconceptions remain from the previous lesson: "Radical Yet Rational, Part 1."

### Explore

15 Minute(s)

Display **slide 6**. Direct students’ attention to the second part of the Search and Solve handout: Solve. Ask students to work in pairs to solve the given equations.

As students work through these problems at their own pace, slowly transition through **slides 7–8** so students can check their work on the first two questions. Use this time to clear up any misconceptions students still have about solving quadratic equations.

Go to **slide 9** and ask students to share their thinking on how to solve questions 3 and 4—what they know, what they are unsure of, etc. Remind the class it is okay that these problems are not completely solved yet, since this time is for students to determine what they already know.

As students share, take note of which misconceptions exist. Be sure to address these during the Explain portion of the lesson.

### Explain

25 Minute(s)

Display **slide 10** and pass out the attached **Guided Notes **handout to each student. Complete the handout as a class. Have students add this handout to their math notebooks if that is a classroom norm.

### Extend

20 Minute(s)

Inform students it is time for them to apply what they have learned. Display **slide 11** and pass out the attached **Real-World Rational Exponents** handout to each pair of students.

Have students work in pairs to answer the questions in each scenario. As students wrap up the first scenario, slowly transition through **slides 12–13** so students can check their work.

As students wrap up the second scenario, slowly transition through **slides 14–15** so students can check their work.

### Evaluate

15 Minute(s)

Use the Choice Board strategy to assess students’ ability to solve equations with radicals and rational exponents using either method. Display **slide 16** and pass out the attached **Choice Board** handout to each student.

The handout contains nine questions, which are divided into three columns (A, B, and C) and three rows (1, 2, and 3). Students are to select one question per column and one question per row, completing a total of three questions.

In column A, students must provide a written explanation of how to solve one of the equations containing rational exponents. In column B, students must algebraically solve one equation containing radicals. In column C, students must algebraically solve one equation containing rational exponents.

Use student responses to see which misconceptions persist before moving on to the next lesson: "Radical Yet Rational, Part 3."

### Resources

ElisaRiva. (2017, February 13). Brain, mind, psychology [Illustration]. Pixabay. https://pixabay.com/illustrations/brain-mind-psychology-idea-drawing-2062057/

K20 Center. (n.d.). Choice Boards. Strategies. https://learn.k20center.ou.edu/strategy/73

K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081