K20 LEARN | My Imaginary Friend, Part 1

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

Online

Materials

Common Cartridge (attached)
History of Imaginary Numbers Infographic (linked)
Desmos account
Student devices with internet access

Hybrid

Materials

Common Cartridge (attached)
History of Imaginary Numbers Infographic (linked)
Desmos account
Student devices with internet access

Hybrid

Engage

10 Minute(s)

Teacher's Note: Desmos Activity Preparation

To use this Desmos Classroom activity, select the following link: "My Imaginary Friend, Part 1: Synchronous." Create an account or sign in under the "Activity Sessions" heading. After you log in, the green "Assign" dropdown button will be active. Click the arrow next to the word "Assign," then select "Single Session Code." After making some setting selections, select "Create Invitation Code" and give the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to https://k20center.ou.edu/externalapps/using -activities/.

For more detailed information about Desmos features and how-to tips, go to https://k20center.ou.edu/externalapps/desmos -home-page/.

To set up the activity’s pacing for students, select "View Dashboard" (next to the session code). In the upper-left corner of your screen, select the icon above the word "Pacing." Desmos Classroom should then prompt you to select the first and last screens that you want students to see. When prompted to set a range, select screens 1 and 3. Select "Restrict to Screens 1–3" to confirm your selection. This allows students to access only screens 1–3 at this time. For more information about teacher pacing, go to https://k20center.ou.edu/externalapps/pacing -activities/.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Introduce the lesson using screens 1–2 of the Desmos activity. Screen 1 displays the lesson’s essential question. Screen 2 identifies the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.

On screen 3, 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.

On the Desmos dashboard, click the orange plus sign to allow students to progress to screen 4. Using the same strategy, 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.

Online

Engage

10 Minute(s)

Teacher's Note: Desmos Activity Preparation

Before starting the activity, assign student pairs. In an asynchronous setting, advise students to schedule a time to meet with their partners.

To use this Desmos Classroom activity, select the following link: "My Imaginary Friend, Part 1: Asynchronous." Create an account or sign in under the "Activity Sessions" heading. After you log in, the green "Assign" dropdown button will be active. Click the arrow next to the word "Assign," then select "Single Session Code." After making some setting selections, select "Create Invitation Code" and give the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to https://k20center.ou.edu/externalapps/using -activities/.

For more detailed information about Desmos features and how-to tips, go to https://k20center.ou.edu/externalapps/desmos -home-page/.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

On screen 1, 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. Students must also provide their reasoning behind the selection. After submitting their answers, students are able to see their peers’ responses.

On screen 2, students decide which of the following radicals is not like the others: the square roots of 20, 75, 16, -16, and 9. Again, students must explain their reasoning.

Screen 3 prompts students to think differently about that same list of radicals. Students must pick a different radical that does not belong and provide justification.

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.

Online

Explore

10 Minute(s)

On screen 4, each student needs to collaborate with their partner. Students start by simplifying the square root of 16. After submitting the correct answer, they are prompted to share their thinking. The response they enter is then shared with their classmates.

Next, students try to simplify the square root of -16 and share their thinking.

On screen 5, students start by simplifying the square root of 20. After submitting the correct answer, they are prompted to share their thinking. The response they enter is shared with their classmates.

Then, students must try again to simplify the square root of -16 and share their thinking.

On screen 6, students use the I Notice, I Wonder strategy to reflect on what they just did. Their submitted responses are then shared with their classmates.

Hybrid

Explore

10 Minute(s)

On the Desmos dashboard, click the orange "Edit" button and select screens 5 and 7. This allows students to access screens 5–7.

On screen 5, have students work in pairs to simplify the square root of 16. After submitting the correct answer, they are prompted to share their thinking.

Next, ask students to try to simplify the square root of -16.

After a few minutes, encourage students to move to screen 6 and simplify the square root of 20. After submitting the correct answer, they are prompted to share their thinking.

Then, ask students to try again to simplify the square root of -16. After a few minutes, have students go to screen 7 and use the I Notice, I Wonder strategy to reflect on what they just did.

Hybrid

Explain

25 Minute(s)

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.

On the Desmos dashboard, click the orange "Edit" button and select screen 8. This screen prompts students to leave the Desmos activity and go to the provided infographic link to read about the history of imaginary numbers. Give students 5 quiet minutes to read about the history of imaginary numbers.

At the end of the infographic, have students watch the "My Imaginary Friend, Part 1" video. The video shows students how to simplify the square roots of -16, -20, and -18.

On the dashboard, click the orange plus sign to allow students to progress to screen 9, then click again for screen 10 as students practice what they’ve just learned.

Students should work in pairs to simplify the square root of -48 and the negative square root of -36. The Desmos activity has a built-in self-check on these screens, so students receive immediate feedback about their responses. After submitting the correct answer, students are asked to explain their thinking, which is then shared with their classmates.

Online

Explain

25 Minute(s)

Screen 7 prompts students to leave the Desmos activity and go to the provided infographic link to read about the history of imaginary numbers.

At the end of the infographic, students should watch the "My Imaginary Friend, Part 1" video. The video shows them how to simplify the square roots of -16, -20, and -18.

On screens 8–9, students practice what they’ve just learned. Students should work with their partners to simplify the square root of -48 and the negative square root of -36.

The Desmos activity has a built-in self-check on these screens, so students receive immediate feedback about their responses. After submitting the correct answer, students are asked to explain their thinking, which is then shared with their classmates.

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.

Teacher's Note: Guiding the Lesson

If students are struggling to see a pattern, suggest that they compare each row or have students try to predict the results of the next column, if the table were to continue. Many of students’ generalizations are likely to come from how they approached completing the table. Guide students to see the benefits of different approaches.

Some students may notice a cyclical pattern; some may notice the last row is always +1. Some may observe that when i is taken to a multiple of four power, i^{(multiple of 4)}, the result is +1. Some students may look at the exponent and write it with as many factors of i² as possible.

For example, a student might write: i²³ = i²²·i = (i²)¹¹·i = (-1)¹¹·i = -i; in that case, a student only needs to remember the math fact of i² = -1.

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.

Online

Extend

20 Minute(s)

On screens 10–12, students must work through a series of tables to discover patterns for simplifying i to a power. The Desmos activity has a built-in self-check on these screens, so students receive immediate feedback about their completed tables.

On screen 13, students see all of their responses collected in one table. Students then reflect on the table, again using the I Notice, I Wonder strategy. Their submitted responses are shared with their classmates.

On screen 14, students apply the patterns they noticed to simplify i¹⁰⁰, i⁴⁵, and i⁶⁷. The Desmos activity has a built-in self-check on this screen, so students receive immediate feedback about their results.

Hybrid

Extend

20 Minute(s)

On the Desmos dashboard, click the orange "Edit" button and select screens 11 and 14. This allows students to access screens 11–14.

Beginning on screen 11, students should work with their partners to complete the tables and then progress to screens 12–13. Encourage students to use properties of exponents and the given information, i¹ = i and i² = -1, to complete the table. The Desmos activity has a built-in self-check on these screens, so students receive immediate feedback about their completed tables.

On screen 14, students see all of their responses collected in one table. Students then reflect on the table, again using the I Notice, I Wonder strategy.

Next, ask students to 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 on screen 14.

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.

Teacher's Note: Guiding the Lesson

If students are struggling to see a pattern, suggest that they compare each row or have students try to predict the results of the next column, if the table were to continue. Many of students’ generalizations are likely to come from how they approached completing the table. Guide students to see the benefits of different approaches.

Some students may notice a cyclical pattern; some may notice the last row is always +1. Some may observe that when i is taken to a multiple of four power, i^{(multiple of 4)}, the result is +1. Some students may look at the exponent and write it with as many factors of i² as possible.

For example, a student might write: i²³ = i²²·i = (i²)¹¹·i = (-1)¹¹·i = -i; in that case, a student only needs to remember the math fact of i² = -1.

On the dashboard, click the orange plus sign to allow students to progress to screen 15. On this screen, students apply the patterns they noticed to larger exponents. Have students work with their partners to simplify i¹⁰⁰, i⁴⁵, and i⁶⁷. The Desmos activity has a built-in self-check on this screen, so students receive immediate feedback about their results.

Hybrid

Evaluate

5 Minute(s)

Use the Always, Sometimes, or Never True strategy to assess what students have learned during the lesson. On the Desmos dashboard, click the orange "Edit" button and select screen 16. This displays screen 16 for students.

Explain to students that they must decide whether "always," "sometimes," or "never" is the most appropriate word to describe how often each of the following statements is true. Students also must be able to explain their reasoning.

As students decide whether each statement is "always," "sometimes," or "never" true, have them sort the statement cards into those three groups by dragging the cards on top of one another.

i is a real number.
i to an even power is -1.
i to an odd power is +i or -i.
i to a multiple of 4 power is +1.
i to a power simplifies to a complex number.
i to a power simplifies to a real number.
i to a power simplifies to an imaginary number.
The square root of a positive real number is an imaginary number.
A real number squared is a negative number.

Use student responses to see which misconceptions persist.

Online

Evaluate

5 Minute(s)

Using the Always, Sometimes, or Never True strategy on screen 15, students reflect and are assessed on what they have learned during the lesson.

Students must decide whether each of the following statements is always true, sometimes true, or never true. They then sort the cards into those three groups by dragging the cards on top of one another.

i is a real number.
i to an even power is -1.
i to an odd power is +i or -i.
i to a multiple of 4 power is +1.
i to a power simplifies to a complex number.
i to a power simplifies to a real number.
i to a power simplifies to an imaginary number.
The square root of a positive real number is an imaginary number.
A real number squared is a negative number.

Use student responses to see which misconceptions persist.

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:

i is a real number.
i to an even power is -1.
i to an odd power is +i or -i.
i to a multiple of 4 power is +1.
i to a power simplifies to a complex number.
i to a power simplifies to a real number.
i to a power simplifies to an imaginary number.

Resources

K20 Center. (n.d.). Always, Sometimes, or Never True. Strategies. https://learn.k20center.ou.edu/strategy/145
K20 Center. (n.d.). I Notice, I Wonder. Strategies.
https://learn.k20center.ou.edu/strategy/180
K20 Center. (n.d.). My Imaginary Friend, Part 1 [Video]. YouTube. https://www.youtube.com/watch?v=pj3S6X-b5Yc
K20 Center. (n.d.). Not Like the Others. Strategies.
https://learn.k20center.ou.edu/strategy/77
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081

Authentic Lessons for 21st Century Learning

My Imaginary Friend, Part 1

Understanding and Simplifying Imaginary Numbers

Summary

Essential Question(s)

Snapshot

Instructional Formats

Materials

Materials

Materials

Engage

Engage

Engage

Explore

Explore

Explore

Explain

Explain

Explain

Extend

Extend

Extend

Evaluate

Evaluate

Evaluate

Resources

Publisher

Downloads

Standards

Facilitator Resources

Learner Handouts

Authentic Lessons for 21st Century Learning

Summary

Essential Question(s)

Snapshot

Instructional Formats

Materials

Materials

Materials

Engage

Engage

Engage

Explore

Explore

Explore

Explain

Explain

Explain

Extend

Extend

Extend

Evaluate

Evaluate

Evaluate

Resources

Publisher

Downloads

Standards

Facilitator Resources

Learner Handouts

Related Resources