### Summary

To complete this lesson, students need to know how to simplify the square root of a negative number and how to simplify i to a power.* This lesson is contained within an escape room. Students will navigate through a spaceship and, along the way, will perform operations on complex numbers so they can make it to the escape pods in time. Students will see the real-world applications of these operations through learning about electrical circuits. Students also will watch a video of an electrical engineer speaking about his career, describing how he uses electrical circuits, and explaining how to divide complex numbers using complex conjugates. *See the "My Imaginary Friend, Part 1" lesson for the prerequisite content. 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)

Why do we perform operations on complex numbers?

### Snapshot

**Engage**

Students read a story that sets the scene for the escape room in a spaceship. Students use their reasoning skills to turn given clues into a passcode as they learn how to navigate through the escape room.

**Explore**

Students are introduced to electrical circuits and perform calculations using electrical circuits. Students add and subtract complex numbers to find the total impedance, then multiply complex numbers to find the voltage.

**Explain**

Students watch a video of an electrical engineer speaking about his career and describing how he uses electrical circuits. Students then learn about complex conjugates and how to use them to divide complex numbers.

**Extend**

Students apply what they have learned about performing operations with complex numbers to progress through the escape room.

**Evaluate**

Students use the Fist to Five strategy to reflect on what they have learned during the lesson.

### 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)

Lesson Guide and Escape Room Script (attached; for teacher use; printed front/back)

User Manual—Definitions handout (attached; one per student; printed front only)

User Manual—Calculations handout (attached; one per student; printed front/back)

User Manual—Calculations (Sample Responses) (attached; for teacher use)

Power Control Hub handout (attached; one per student; printed front only)

Cargo Bay handout (attached; one per student; printed front only)

Pencils

Student devices with internet access**

Go With the Flow handout (optional; attached; one half-sheet per student; printed front only)

Calibration Dial handout (optional; attached; creates 2 dials per document; printed front only)

**See the note in the Extend portion of this lesson for alternatives to having student devices.

### Engage

5 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.

Display **slide 5** and read aloud the directions to the class. Explain to students that they need to assemble a crew to escape successfully. Have students work in pairs throughout the lesson.

Display **slide 6**, which begins the escape room story. Read this screen aloud to the class. Transition through **slides 7–8** and continue reading the story aloud.

Move to **slide 9** and have students work with their partners to determine the three-letter passcode.

Give students a couple minutes to figure out the passcode before moving to the next slide. Display **slide 10** to show students the correct passcode.

### Explore

20 Minute(s)

Transition through **slides 11–12** and continue reading the story aloud. Display **slide 13** and pass out the attached **User Manual—Definitions** handout. Give students a few minutes to read through the handout.

Move to **slide 14** and direct students’ attention to example 1 at the bottom of the handout. Then, transition through **slides 15–17** to help students understand how to calculate the total impedance of an electrical circuit.

Next, display **slide 18**. Have students work with their partners to find the total impedance of example 2, which is also located at the bottom of the User Manual—Definitions handout.

While students discuss, pass out the attached **User Manual—Calculations** handout. Then, ask for volunteers to explain why the total impedance is (30+3*i*) ohms.

Once students understand the two example problems, display **slide 19**. Ask students to work with their partners to find the total impedance for questions 1 and 2 on the User Manual—Calculations handout.

Display **slide 22** and have students work with their partners to find the voltage for questions 3 and 4 on the handout.

### Explain

20 Minute(s)

Display **slide 25** and have students try to answer question 5 from the User Manual—Calculations handout. Give students a few minutes to struggle with the problem. Then, ask the class what operation they need to use with complex numbers in this scenario. Ask for volunteers to explain why they need to divide complex numbers.

Move to **slide 26**. Introduce the "My Imaginary Friend, Part 2" video, which features an electrical engineer talking about his career, describing how he uses complex numbers, and explaining how to divide complex numbers. Play the video.

The video begins with the electrical engineer interview, which ends at the 12:45 mark. The rest of the video demonstrates how to divide complex numbers using a complex conjugate.

After the video, display **slide 27** so students can see the entire worked-out solution for question 5. Ask students to discuss the problem with their partners and ask one another questions.

As a class, ask students what questions they still have. After addressing any questions, move to **slide 28** and have students find the current for question 6 on the handout.

### Extend

30 Minute(s)

Transition through **slides 30–31** and continue reading the story aloud. Then, display **slide 32** and pass out the attached **Power Control Hub** handout. Remind students to read everything on the handout before they begin working on it with their partners.

While students work, monitor student progress while walking around the room with the Lesson Guide and Escape Room Script on hand. Be sure not to immediately check answers or give assistance. It is important for students to have a little time to struggle as they work through these challenging problems.

As students start the third question on the handout, consider staying at your desk or a location in which students can easily and comfortably reach you to check their three-letter code. As noted in the Lesson Guide and Escape Room Script, if students have the correct Transfer Power Code, then you should exchange students’ Power Control Hub handout for the attached **Cargo Bay** handout.

Students may finish these handouts at staggering times, so having quick reference to the Lesson Guide and Escape Room Script can help you ensure the activity goes smoothly. When students finish the Cargo Bay handout and come to you to check their Release Code, let students who are correct know that they’ve escaped successfully (and that they must keep the code a secret).

Use the Power Control Hub and Cargo Bay handouts to determine what misconceptions students may have regarding operations with complex numbers.

Once the whole class has escaped successfully, display **slide 33**.

### Evaluate

5 Minute(s)

Display **slide 34** and have students use the Fist to Five strategy to reflect on what they’ve learned during the lesson. On slide 34, ask students to focus their reflection on their understanding of adding and subtracting complex numbers.

Then, display **slide 35** and have students reflect on their understanding of multiplying complex numbers. Finally, display **slide 36** and have students reflect on their understanding of dividing complex numbers.

### Resources

K20 Center. (n.d.). Fist to Five. Strategies. https://learn.k20center.ou.edu/strategy/68

K20 Center. (n.d.). GeoGebra. Tech tools. https://learn.k20center.ou.edu/tech-tool/2352

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

K20 Center. (n.d.). My Imaginary Friend, Part 2 cover image [Composite image]. K20 Center.

Vecteezy. (n.d.). Rocket Ship Flying [Image]. Vecteezy. https://www.vecteezy.com/vector-art/1871713-happy-successful-businessman-holding-goal-flag-standing-on-rocket-ship-flying-through-starry-sky-business-startup-concept

Vecteezy. (n.d.). Round Spaceship on White Background [Image]. Vecteezy. https://www.vecteezy.com/vector-art/359100-round-spaceship-on-white-background