K20 LEARN | My Imaginary Friend, Part 2

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.

Face-to-Face

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.

Online

Materials

Common Cartridge (attached)
Desmos account
Student devices with internet access
Paper
Pencils

Hybrid

Materials

Common Cartridge (attached)
User Manual—Definitions handout (attached; one per student; printed front only)
Desmos account
Student devices with internet access
Paper
Pencils
Lesson Guide and Escape Room Script (optional; attached; for teacher use)
User Manual—Calculations (Sample Responses) (optional; attached; for teacher use)

Hybrid

Engage

5 Minute(s)

Teacher's Note: Desmos Activity Preparation

To use this Desmos Classroom activity, select the following link: "My Imaginary Friend, Part 2: 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 the Desmos dashboard, click the orange "Edit" button and select screens 3 and 7. This allows students to access screens 3–7. Beginning on screen 3, have students work in pairs to progress to the next check-in point, which is screen 7.

The Desmos activity’s design requires students to enter the correct answer(s) on each screen in order to progress to the next. For example, if students do not enter their names on screen 3, then screens 4–6 show a message that says "Access Denied," with directions to go back to enter the names of their crew.

Students must solve their first riddle on screen 6, where they are given a circuit diagram, a list of symbols with corresponding letters, and a clue that says, "Go with the flow." Students are expected to use the three symbols in the diagram, find the corresponding letters, and enter those three letters in the table to unlock the next screen. Later in the lesson, students learn what a circuit diagram is and what those symbols mean.

If students enter the code backwards, they see an error message (orange triangle) with a hint to go the opposite direction. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter. Once students enter the correct three-letter code, they can progress to the next screen.

Check the Desmos dashboard to view student progress. As students reach screen 7, pass out the attached User Manual—Definitions handout.

Online

Engage

5 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 2: 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.

Students must solve their first riddle on screen 4, where they are given a circuit diagram, a list of symbols with corresponding letters, and a clue that says, "Go with the flow." Students are expected to use the three symbols in the diagram, find the corresponding letters, and enter those three letters in the table to unlock the next screen. Later in the lesson, students learn what a circuit diagram is and what those symbols mean.

If students enter the code backwards, they see an error message (orange triangle) with a hint to go the opposite direction. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter. Once students enter the correct three-letter code, they can progress to the next screen.

Face-to-Face

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.

Face-to-Face

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+3i) 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.

Online

Explore

20 Minute(s)

On screen 5, students continue to read the story and learn that they need to read the user manual. On screens 6–9, students read the user manual, which contains definitions and examples regarding electrical circuits. Learning about computations with electrical circuits is a real-world application of operations with complex numbers.

On screen 6, students learn about circuits, current, and voltage. On screen 7, students learn about impedance and circuit components: source, resistor, inductor, and capacitor.

On screen 8, students walk through an example of how to calculate the total impedance of a circuit. Going through this example unlocks the next screen. On screen 9, students see a second, more complicated example of calculating total impedance. Again, going through this example unlocks the following screen.

Students on screen 10 are asked to calculate the total impedance of a circuit on their own. Submitting the correct answer unlocks the next screen. On screen 11, students must find the total impedance of a second circuit. Again, submitting the correct answer unlocks the following screen.

On screens 12–13, students are given the voltage formula: Voltage = (Current)·(Impedance). Students then are asked to calculate the voltage, given the current and impedance. Submitting a correct answer on each screen unlocks the next.

Hybrid

Explore

20 Minute(s)

On the Desmos dashboard, click the orange "Edit" button and select screens 3 and 14. This allows students to access screens 3–14. Have students work in pairs to progress to the next check-in point, which is screen 14. Students may find it helpful or even necessary to review previous screens as they work, so be careful to ensure the pacing does not limit students’ ability to look at previous screens.

On screen 8, students are asked to read the User Manual—Definitions handout, which contains definitions and examples regarding electrical circuits. Learning about computations with electrical circuits is a real-world application of operations with complex numbers.

On screens 9–10, students walk through examples 1 and 2 from the handout with a more thorough explanation of how to calculate the total impedance of a circuit. Going through each example unlocks the following screens.

Students on screen 11 are asked to calculate the total impedance of a circuit on their own. Submitting the correct answer unlocks the next screen. On screen 12, students must find the total impedance of a second circuit. Again, submitting the correct answer unlocks the following screen.

On screens 13–14, students are given the voltage formula: Voltage = (Current)·(Impedance). Students then are asked to calculate the voltage, given the current and impedance. Submitting a correct answer on each screen unlocks the next.

Hybrid

Explain

20 Minute(s)

Teacher's Note: Lesson Pacing

This lesson can be naturally paused and resumed at different points during the Explain portion, depending on how the class is progressing. Ideally, pause the lesson at the end of the Explain portion and resume the next day with the Extend portion.

You may find students need more time due to misconceptions with prerequisite knowledge. In this case, consider pausing the lesson after the video and question 5 and resuming the next day with question 6. You also could have students watch the video and do question 5 at home and resume the next day with question 6 in class. Alternatively, consider allowing the lesson to take three 45-minute class periods, as follows: Day 1, Engage & Explore; Day 2, Explain; and Day 3, Extend & Evaluate.

You do not actually need to pause the student pacing in Desmos.

On the Desmos dashboard, click the orange plus sign to allow students to progress to screen 15. On this screen, students are asked to use the voltage formula to calculate the current, given the voltage and impedance. This calculation requires students to divide complex numbers, which likely is not something they have done before.

In the upper-left corner of your screen, click the icon above the word "Pause." 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.

After the class discussion, introduce the "My Imaginary Friend, Part 2" video, which features an electrical engineer talking about his career, describing how he uses electrical circuits, and explaining how to divide complex numbers. Direct students’ attention to the classroom screen and 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.

Alternatively, you may have students watch the video on their own through Desmos. In this case, ask students to use the link on screen 15. The question on screen 15 is the example provided in the video.

To allow students to resume the Desmos activity, click the icon above the word "Pause." Watching the video provides students with the correct answer to submit, which unlocks the following screen.

On the Desmos dashboard, click the orange plus sign to allow students to progress to screen 16. On this screen, students are asked to calculate the current on their own. Again, submitting a correct answer unlocks the next screen.

Online

Explain

20 Minute(s)

On screen 14, students are asked to use the voltage formula to calculate the current, given the voltage and impedance. This calculation requires students to divide complex numbers, which likely is not something they have done before.

To aid them in dividing complex numbers, students are asked to use the link on screen 14 to watch the "My Imaginary Friend, Part 2" video, which features an electrical engineer talking about his career, describing how he uses electrical circuits, and explaining how to divide complex numbers. The question on screen 14 is the example provided in 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.

Watching the video provides students with the correct answer to the question on screen 14, and submitting it unlocks the following screen. On screen 15, students are asked to calculate the current on their own. Again, submitting a correct answer unlocks the next screen.

Face-to-Face

Explain

20 Minute(s)

Teacher's Note: Lesson Pacing

This lesson can be naturally paused and resumed at different points during the Explain portion, depending on how the class is progressing. Ideally, pause the lesson at the end of the Explain portion and resume the lesson the next day with the Extend portion.

You may find students need more time due to misconceptions with prerequisite knowledge. In this case, consider pausing the lesson after the video and question 5 and resuming the next day with question 6. You also could have students watch the video and do question 5 at home, then resume the next day with question 6 in class—just be sure to provide students with the link to the video.

Alternatively, consider allowing the lesson to take three 45-minute class periods, as follows: Day 1, Engage & Explore; Day 2, Explain; and Day 3, Extend & Evaluate.

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.

Face-to-Face

Extend

30 Minute(s)

Technology Alternatives and Optional Handout

This is the portion where each pair of students needs one device with internet access, as there are two GeoGebra activities that students must navigate through. However, if technology is limited, the escape room still can be completed with the use of a SMART Board to display the webpage for the Cargo Bay handout.

The calibration dial for the Power Control Hub handout can be physically constructed before class using the attached Calibration Dial handout. This is a two-page document that ideally should be printed on cardstock or heavier paper and could be reused. To ensure students have all necessary layers for the calibration dial, each student pair should cut out a half-sheet from the first page and a half-sheet from the second page of the document. Constructing the dial also requires a brad or other paper fastener—even a paper clip could work. Directions for how to cut and assemble the calibration dial are included in the document.

This portion also could be done with stations if only a few online devices are available and not enough for each pair to have access. If this is the case, consider passing out the Cargo Bay handout to half the class while the other half works on the Power Control Hub handout to help with any potential congestion at the stations.

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.

Online

Extend

30 Minute(s)

On screen 16, students continue to read the story. Screen 17 prompts students to leave the Desmos activity and go to the provided link to the “Power Control Hub” GeoGebra activity: geogebra.org/m/rwwffvkr.

In this activity, students must answer a series of questions about complex conjugates, dividing complex numbers, and simplifying radicals. Students should put their answers in the calibration dial (modified cipher wheel) within the GeoGebra activity to find a three-letter code.

Once they have the three-letter code, students should return to the Desmos activity to enter the code and unlock the next screen. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter.

On screen 18, students continue reading the story. Screen 19 prompts students to leave the Desmos activity and go to the provided link to the “Cargo Bay” GeoGebra activity: geogebra.org/m/whtn2brj.

In this activity, students must answer a series of questions about adding and subtracting complex numbers, multiplying complex numbers, and simplifying i to a power. Using the given flashlight tool, students should input their answers to find a three-letter code.

Once they have the three-letter code, students should return to the Desmos activity to enter the code and unlock the next screen. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter.

On screen 20, students read the conclusion of their escape room story and learn they have escaped successfully.

Hybrid

Extend

30 Minute(s)

On the Desmos dashboard, click the orange "Stop" button to allow students to progress through the rest of the screens at their own pace.

As students work with their partners, monitor student progress while walking around the room. 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.

On screen 17, students continue to read the story. Screen 18 prompts students to leave the Desmos activity and go to the provided link to the “Power Control Hub” GeoGebra activity: geogebra.org/m/rwwffvkr.

In this activity, students must answer a series of questions about complex conjugates, dividing complex numbers, and simplifying radicals. Students should put their answers in the calibration dial (modified cipher wheel) within the GeoGebra activity to find a three-letter code.

Once they have the three-letter code, students should return to the Desmos activity to enter the code and unlock the next screen. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter.

On screen 19, students continue reading the story. Screen 20 prompts students to leave the Desmos activity and go to the provided link to the “Cargo Bay” GeoGebra activity: geogebra.org/m/whtn2brj.

In this activity, students must answer a series of questions about adding and subtracting complex numbers, multiplying complex numbers, and simplifying to a power. Using the given flashlight tool, students should input their answers to find a three-letter code.

Once they have the three-letter code, students should return to the Desmos activity to enter the code and unlock the next screen. If students enter a correct letter but do not capitalize the letter, they see an error message with a hint to capitalize the letter.

On screen 21, students read the conclusion of their escape room story and learn they have escaped successfully.

Hybrid

Evaluate

5 Minute(s)

On screen 22, students are asked to use a modified version of the Fist to Five strategy to reflect on what they’ve learned during the lesson. The screen shows a thumbs-down icon and a thumbs-up icon at the ends of a scale. Students should move a point along the scale to show how confident they feel about each of the operations on complex numbers: addition and subtraction, multiplication, and division.

Students also have the option to type and elaborate on how they feel. There is not a submit button on this screen since the written responses are optional.

Online

Evaluate

5 Minute(s)

On screen 21, students are asked to use a modified version of the Fist to Five strategy to reflect on what they’ve learned during the lesson. The screen shows a thumbs-down icon and a thumbs-up icon at the ends of a scale. Students should move a point along the scale to show how confident they feel about each of the operations on complex numbers: addition and subtraction, multiplication, and division.

Students also have the option to type and elaborate on how they feel. There is not a submit button on this screen since the written responses are optional.

Face-to-Face

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

Authentic Lessons for 21st Century Learning

My Imaginary Friend, Part 2

Operations with Complex 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