Lesson 14

More Arithmetic with Complex Numbers

14.1: Which One Doesn’t Belong: Complex Expressions (5 minutes)

Warm-up

This warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Which one doesn’t belong?

A. \(i^2\)

B. \((1 + i) + (1 - i)\)

C. \((1 + i)^2\)

D. \((1 + i)(1 - i)\)

Student Response

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Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as real part or imaginary part. Also, press students on unsubstantiated claims.

14.2: Powers of $i$ (15 minutes)

Optional activity

This activity is optional because it goes beyond the depth of understanding required to address the standard.

In this activity, students use repeated reasoning to find patterns in the powers of \(i\). Students find that powers of \(i\) correspond to a repeating sequence \(1, i, \text-1, \text-i\).

Monitor for students who:

  • Break the expressions into repeated factors of \(i^2\)
  • Discover and use the fact that \(i^4=1\) to reduce the exponent to the remainder upon division by 4
  • Use the exponent rule \(x^a \boldcdot x^b = x^{a+b}\) to calculate
  • Use the exponent rule \((x^a)^b=x^{ab}\) to calculate

Launch

Arrange students in groups of 2. Encourage students to check with their partner about each power of \(i\) before going on to the next power. Follow with a whole-class discussion.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written explanations for finding the values of \(i^{100}\) and \(i^{38}\). Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the value of \(i^4\)?”, “Which exponent rule did you use?”, “How did you use the exponent rule to find the value of \(i^{100}\)?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students explain their strategy for calculating powers of \(i\).
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

  1. Write each power of \(i\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. If \(a\) or \(b\) is zero, you can ignore that part of the number. For example, \(0+3i\) can simply be expressed as \(3i\).

    \(i^0\)

    \(i^1\)

    \(i^2\)

    \(i^3\)

    \(i^4\)

    \(i^5\)

    \(i^6\)

    \(i^7\)

    \(i^8\)

  2. What is \(i^{100}\)? Explain your reasoning.
  3. What is \(i^{38}\)? Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

  1. Write each power of \(1+i\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. If \(a\) or \(b\) is zero, you can ignore that part of the number. For example, \(0+3i\) can simply be expressed as \(3i\).
    1. \((1+i)^0\)
    2. \((1+i)^1\)
    3. \((1+i)^2\)
    4. \((1+i)^3\)
    5. \((1+i)^4\)
    6. \((1+i)^5\)
    7. \((1+i)^6\)
    8. \((1+i)^7\)
    9. \((1+i)^8\)
  2. Compare and contrast the powers of \(1+i\) with the powers of \(i\). What is the same? What is different?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select previously identified students to share the patterns they found in this order:

  • Break the expressions into repeated factors of \(i^2\)
  • Discover and use the fact that \(i^4=1\) to reduce the exponent to the remainder upon division by 4
  • Use the exponent rule \(x^a \boldcdot x^b = x^{a+b}\) to calculate
  • Use the exponent rule \((x^a)^b=x^{ab}\) to calculate

Ask students to compare how well these strategies work when the exponent is very large, like with the last two problems. Students might find that combining exponent rules with the fact that \(i^2=\text-1\) and \(i^4=1\) makes it more efficient to compute powers of \(i\) with very large exponents. Close by asking students to compute \(i\) to a very large power, like the calendar year. For example, \(i^{2018}=i^{2000}\boldcdot i^{16} \boldcdot i^2 = (i^4)^{500} \boldcdot (i^4)^4 \boldcdot i^2 = 1 \boldcdot 1 \boldcdot \text-1 = \text-1\).

Action and Expression: Internalize Executive Functions. Provide students with a two-column table to record responses and identify a pattern. Use the left column to list each given power of \(i\) and the right column to list expressions in the form \(a+bi\) for students to identify patterns.
Supports accessibility for: Organization; Attention

14.3: Add 'Em Up (or Subtract or Multiply) (15 minutes)

Optional activity

This activity is optional because it is an opportunity for extra practice that not all classes may need.

The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3). In a row game, pairs of students do different problems that have the same answer. If there are discrepancies in their answers, students must communicate with each other to resolve those discrepancies.

Look for groups who disagree but then work to reach agreement to share during the synthesis. Also look for any groups who initially agree, but upon deeper inspection, were both incorrect.

Launch

Arrange students in groups of 2, assigning one student as partner A and the other as partner B. Explain to students that there will be two columns of problems and that they only do the problems in their column. Students are to complete the problems, and then compare answers with their partner. If they do not get the same answer, they should work together to find the error.

Speaking: MLR8 Discussion Supports. To help students explain their strategy for simplifying the expression, provide sentence frames such as: “First, I _____ because _____. Then, I…”, “I tried _____ and what happened was…”, “How did you get…?”, and “How do you know…?”
Design Principle(s): Optimize output (for explanation)
Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite pairs to agree on 4 of the 6 rows to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

For each row, your partner and you will each rewrite an expression so it has the form \(a+bi\), where \(a\) and \(b\) are real numbers. You and your partner should get the same answer. If you disagree, work to reach agreement.

partner A partner B
\((7 + 9i) + (3 - 4i)\) \(5i(1 - 2i)\)
\(2i(3 + 4i)\) \((1 + 2i) - (9 - 4i)\)
\((4 - 3i)(4 + 3i)\) \((5 + i) + (20 - i)\)
\((2i)^4\) \((3 + i \sqrt{7})(3 - i \sqrt{7})\)
\((1 + i \sqrt{5}) - (\text- 7 - i \sqrt{5})\) \((\text- 2i)(\text- \sqrt{5} + 4i) \)
\(\left( \frac12 i \right) \left( \frac13 i \right) \left( \frac34 i \right)\) \(\left( \frac12 i \right)^3\)

 

Student Response

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Anticipated Misconceptions

In the last problem, students may stop with the answer \(\frac18 i^3\). Remind these students of their work in the previous activity.

Activity Synthesis

Select previously identified groups to share how they resolved disagreement, or how they figured out they were both incorrect. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

Lesson Synthesis

Lesson Synthesis

In this lesson, students practiced arithmetic with complex numbers. Here are some questions for discussion:

  • “How is arithmetic with complex numbers the same as with real numbers? How is it different?” (The arithmetic is the same as with real numbers because it still involves multiplying, exponentiating, adding, and subtracting. It is different because we didn’t divide complex numbers, and also there is an extra consideration that \(i^2=\text-1\).)
  • “What are some things about complex numbers or radicals you didn’t understand very well at first, but now you feel you understand much better?” (I didn’t understand the difference between the \(\sqrt{}\) symbol and its connection to the solutions of a quadratic equation like \(x^2=5\). I now know from graphs that \(x^2=5\) has two solutions, a positive and a negative. The positive solution is written as \(\sqrt{5}\) and the negative solution is written as \(\text-\sqrt{5}\).)

14.4: Cool-down - Operate on Complex Numbers (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Suppose we want to write the product \((3+5i)(7-2i)\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. For example, we might want to compare our solution with a partner’s, and having answers in the same form makes that easier. Using the distributive property,

\(\displaystyle \begin{align} (3+5i)(7-2i) &= 21 - 6i +35i - 10i^2 \\ &= 21 + 29i - 10(\text-1) \\ &= 21+ 29i +10 \\ &= 31 +29i \end{align}\)

Keeping track of the negative signs is especially important since it is easy to mix up the fact that \(i^2=\text-1\) with the fact that \(\text-i^2=\text-(\text-1) = 1\).

Next, suppose we want to write the difference \((\text-6+3i) - (2-4i)\) as a single complex number in the form \(a+bi\). Distributing the negative and combining like terms, we get:

\(\displaystyle \begin{align} (\text-6+3i) - (2-4i) &= \text-6 +3i - 2 - (\text-4i) \\ &= \text-8 +3i +4i \\ &= \text-8 +7i \end{align}\)

Again, it is important to be precise with negative signs. It is a common mistake to just subtract \(4i\) rather than subtracting \(\text-4i\).