Complex Numbers: The Cycle of i and Complex Arithmetic

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Mathematics · Algebra II
SP-ALG2-004 · Free to print and use in your classroom · substituteplans.com
Page 2
Complex Numbers: The Cycle of i and Complex
Arithmetic
Mathematics · Algebra II · Grades 11-12 · 50 minutes
Time needed
50 minutes (one class period)
Prep
Copy the 2-page worksheet (one per student). Write the definition of i and the worked
examples on the board.
Technology
None.
Student
independence
High. Pattern-finding, then individual practice.
Copies to make
One 2-page worksheet per student
Materials
Pencils, Worksheet, Whiteboard or chalkboard
Goal for today: Students will simplify powers of i and will add, subtract, multiply and divide complex numbers written
in the form a + bi.
Before the bell
☐ Copy the worksheet, one per student (2 pages).
☐ Write on the board: i = √−1 and i² = −1. A complex number looks like a + bi, where a and b are real
numbers.
☐ Write worked examples: (2 + 3i) + (1 − 5i) = 3 − 2i. (1 + i)(2 + i) = 2 + i + 2i + i² = 2 + 3i − 1 = 1 + 3i.
☐ Write the conjugate rule: the conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b².
Say this to start
Good morning! Today students work with a special number called i, whose square is −1. It sounds strange, but it
follows clear rules, and the rules are on the board. They will look for a pattern first, then do some arithmetic with these
numbers. Everything they need is on the board and the worksheet.

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Mathematics · Algebra II
SP-ALG2-004 · Free to print and use in your classroom · substituteplans.com
Page 3
Run of show
Minutes
What you do
What students do
0–5
Take attendance. Hand out the worksheet.
Write their names and look over the
worksheet.
5–12
Read the definition and worked examples on the board
aloud. Slowly say: “i squared equals negative one.”
Follow along. Ask questions.
12–20
Tell students to fill in the powers-of-i table. Ask them to
look for a repeating pattern.
Fill in i¹ through i⁸ and describe the pattern
to a neighbor.
20–38
Tell students to complete questions 1–10 in the work
boxes. Walk the room. Point stuck students to the worked
examples.
Simplify powers of i, then add, subtract,
multiply and divide complex numbers.
38–45
Tell students to answer question 11 and true/false
questions 12–14. Then read answers aloud from the key.
Answer the questions and check their work.
45–50
Tell students to write the exit ticket (question 15), then
collect worksheets.
Answer question 15 and hand in the
worksheet.
If things go wrong
• If students say, “Negative numbers can’t have square roots,” tell them that is true for real numbers, which is
why mathematicians created i.
• If students are stuck on powers like i²⁷, tell them to divide the exponent by 4 and use the remainder.
• If a student multiplies binomials incorrectly, point to the worked example and tell them to multiply all four
pairs.
• If a student finishes early, use the early finisher list.
Early finishers
• Compute i⁵⁰, i⁶³ and i²⁰²⁰ using the remainder trick.
• Make up a complex number multiplication whose answer is a real number. Prove it with your work.
Support and challenge
• Support: Let students use the completed powers table to answer questions 1–3 and complete questions 4–8
first.
• Challenge: Ask students to compute (2 + i)³ by multiplying twice. (Answer: 2 + 11i.)
Notes for the teacher (you fill this in)
Jot a few lines before you leave. A full Daily Sub Report is in the free Sub Survival Kit at SubstitutePlans.com.
How far the class got / what to pick up next:
Students who were especially helpful:

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Mathematics · Algebra II
SP-ALG2-004 · Free to print and use in your classroom · substituteplans.com
Page 4
Anything the teacher should know: