My Imaginary Friend, Part 1

Understanding and Simplifying Imaginary Numbers

Michell Eike | Published: May 18th, 2022 by K20 Center

• Subject Mathematics
• Course Algebra 2
• Time Frame 70 minutes
• Duration 2 class periods

Summary

In this lesson, students will be introduced to imaginary numbers and their history. Students will learn how to simplify the square root of a negative number and how to simplify i to a power. Before beginning this lesson, students need to (1) know how to simplify the square root of a whole number and (2) know the Product of Powers and the Power of a Power properties of exponents. This is a multimodality lesson, which means it includes face-to-face, online, and hybrid versions of the lesson. The attachments also include a downloadable Common Cartridge file, which can be imported into a Learning Management System (LMS) such as Canvas or eKadence. The cartridge includes interactive student activities and teacher's notes.

Essential Question(s)

What are imaginary numbers?

Snapshot

Engage

Students use the Not Like the Others strategy to get them in the mindset of finding patterns.

Explore

Students recall what they know about simplifying square roots and apply what they know to attempt simplifying the square root of a negative number.

Explain

Students read about the history of imaginary numbers and learn the formal definition. Students then learn how to simplify the square root of a negative number.

Extend

Students discover the pattern of simplifying i to a power and apply that pattern to larger exponents.

Evaluate

Students sort statements about real, imaginary, and complex numbers into groups using the Always, Sometimes, or Never True strategy.

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.

Face-to-Face

Materials

• Lesson Slides (attached)

• Complex Numbers handout (attached; one per student; printed front/back)

• Complex Numbers (Sample Responses) (attached; for teacher use)

• History of Imaginary Numbers Infographic (attached; one per student; printed front only)

• Pencils

• Paper

Face-to-Face

Engage

10 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 to share the lesson's essential question. Display slide 4 to go over the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Go to slide 5 and pass out the Complex Numbers handout. Inform students they are going to use this handout throughout the lesson. First, have students use the Not Like the Others strategy to decide which item is not like the others, given a picture of a sandwich, a button, an orange slice, and a donut. Then, have students volunteer to share what they selected and why they made their selection.

Using the same strategy, go to slide 6 and have students decide which of the following radicals is not like the others: the square roots of 20, 75, 16, -16, and 9. Again, ask students to share what they selected and why they made their selection.

Face-to-Face

Explore

10 Minute(s)

Display slide 7. Using the Complex Numbers handout, have students work in pairs to simplify the square root of 16 and justify their answer. Next, ask students to try to simplify the square root of -16.

After a few minutes, encourage students to move forward and simplify the square root of 20, justifying their answer. Then, ask students to try again to simplify the square root of -16.

Go to slide 8. Have students use the I Notice, I Wonder strategy to reflect on what they just did. Ask students to write what they notice and what they wonder on the handout.

Face-to-Face

Explain

25 Minute(s)

Display slide 9. As a class, invite students to discuss what they noticed and wondered in the Explore portion. Guide students to come to an agreement that there is not a real number squared that equals a negative number. Display slide 10 to help resolve any misunderstandings about simplifying square roots.

Go to slide 11. Explain to students that there are more numbers than just real numbers. Pass out the attached History of Imaginary Numbers Infographic and give students 5 quiet minutes to read it.

Once students understand why imaginary numbers exist, go to slide 12 and demonstrate how to simplify the square root of a negative number.

Display slide 14 and have students practice what they've just learned. Ask students to work with their partners to simplify the square root of -48 and the negative square root of -36 on the Complex Numbers handout.

Face-to-Face

Extend

20 Minute(s)

Display slide 15 and have students work with their partners to complete the table on the handout. Encourage students to use properties of exponents and the given information, i1 = i and i2 = -1, to complete the table.

Show slide 16 and have students check their tables. Be sure to allow students time to find their mistakes and refine their understanding.

Display slide 17. Ask students to write what they notice and wonder regarding the table on the handout.

After students are done writing, have them work with their partners to generalize a pattern and develop "rules" or observations they could use to quickly simplify i to a much larger power. They may also develop and record their ideas in the Extend: I Notice, I Wonder portion of the handout.

Display slide 18. As a class, have students volunteer to share what they noticed and wondered. Use these observations and questions to develop strategies for simplifying i to any whole number power.

Display slide 19 and have students apply the patterns they noticed to larger exponents. Ask students to work with their partners to simplify i100, i45, and i67 on the handout.

When students are done, display slide 20 so students can check their work. Be sure to allow students time to find their mistakes and refine their understanding.

Face-to-Face

Evaluate

5 Minute(s)

Display slide 21. Use the Always, Sometimes, or Never True strategy to assess what students have learned during the lesson. Explain to students that they must decide whether "always," "sometimes," or "never" is the most appropriate word to describe how often each statement on the following slide is true. Students also must be able to explain their reasoning.

Show slide 22 and have students decide whether each statement is "always," "sometimes," or "never" true:

1. i is a real number.

2. i to an even power is -1.

3. i to an odd power is +i or -i.

4. i to a multiple of 4 power is +1.

5. i to a power simplifies to a complex number.

6. i to a power simplifies to a real number.

7. i to a power simplifies to an imaginary number.