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Unit 4, Lesson 12

Arithmetic with Complex Numbers

Algebra 2

12.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR5: Co-Craft Questions

Materials

None

Activity Narrative

In this activity, students examine an image of the complex plane showing a path of arrows. They craft questions based on the path that will be addressed later in this lesson.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of the complex plane. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.

Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before comparing questions with a partner.

Activity

None

<p>Line segments drawn on coordinate plane. Starting at origin, right to 2 comma 0, up to 2 comma 3i, left to -6 comma 3i, down to -6 comma -5i.</p>

Student Response

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Building on Student Thinking

Activity Synthesis

Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “complex plane,” “sum,” and “complex number.”

12.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this activity is to introduce the idea that sums of complex numbers can be rewritten as a single complex number in the form , where and are real numbers. In previous lessons, students have used the complex plane to visualize complex numbers, and now they continue that work using the complex plane to visualize why the sum of two complex numbers is another complex number.

12.3

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity, students compare what happens on the real number line and on the complex plane when multiplying numbers. This reinforces the idea that the complex plane is a way of representing complex numbers, analogous to the real number line, and not like the coordinate plane, which represents pairs of numbers.

Launch

None

Activity

None

Lesson Synthesis

Display for all to see the prompt "How are complex numbers and real numbers similar? How are they different?"

Ask students to record their thoughts in writing. After 2 minutes of quiet work time, invite students to name a similarity or a difference that they noticed. Write down student responses for all to see. Press students to be specific in their answers, and help them refine their thoughts if necessary. In particular, if students say that both complex numbers and real numbers can be graphed, ask them to be clear about what they mean. Real numbers can be graphed on a number line and complex numbers can be graphed on the complex plane, but this is not the same kind of graph as the coordinate plane.

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Student Lesson Summary

When we add a real number with an imaginary number, we get a complex number. We usually write complex numbers as:

where and are real numbers. We say that is the “real part” and is the “imaginary part.”

To add (or subtract) two complex numbers, we add (or subtract) the real parts and add (or subtract) the imaginary parts. For example:

In general:

and:

When we raise an imaginary number to a power, we can use the fact that to write the result in the form . For example, . We can group the factors together to see how to rewrite this.

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Audience

Assessment Access

Lesson 12

Arithmetic with Complex Numbers

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSN-CN.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSN-CN.A.2

Building Toward

HSN-CN.A.1

Standards Alignment

Building On

Addressing

HSN-CN.A.2

Building Toward

HSN-CN.A.1