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
For access, consult one of our IM Certified Partners.
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.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
-
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 \(3i\).
\(i^0\)
\(i^1\)
\(i^2\)
\(i^3\)
\(i^4\)
\(i^5\)
\(i^6\)
\(i^7\)
\(i^8\)
- What is \(i^{100}\)? Explain your reasoning.
- 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?
-
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 \(3i\).
- \((1+i)^0\)
- \((1+i)^1\)
- \((1+i)^2\)
- \((1+i)^3\)
- \((1+i)^4\)
- \((1+i)^5\)
- \((1+i)^6\)
- \((1+i)^7\)
- \((1+i)^8\)
- 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, \(i^{2018}=i^{2000}\boldcdot i^{16} \boldcdot i^2 = (i^4)^{500} \boldcdot (i^4)^4 \boldcdot i^2 = 1 \boldcdot 1 \boldcdot \text-1 = \text-1\).
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.
Design Principle(s): Optimize output (for explanation)
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
For access, consult one of our IM Certified Partners.
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
For access, consult one of our IM Certified Partners.
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 \(\text-4i\).