Algebra II Lesson

Complex Numbers: From Imaginary Roots to the Complex Plane

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11th grade Honors — Level III (Honors) — Level III (Honors)BTC Style / Thinking Classrooms90 min

Learning Objective

Students will define the imaginary unit $i = - 1$ , simplify powers of $i$ , perform addition, subtraction, multiplication, and division of complex numbers, and solve quadratic equations with non-real roots using the quadratic formula.

Lesson Overview

Students discover the imaginary unit by encountering a quadratic with no real roots, then build fluency with complex number arithmetic and solving quadratics with non-real solutions using the quadratic formula.

Materials

Vertical whiteboards or large chart paper
Dry-erase markers (multiple colors)
Pencils and paper for notes
Basic calculator (phone calculator is fine)

Scaffolded Task Progression

1Task 1

Evaluate: what real number

x

satisfies

x^{2} = - 1

? Discuss with your group. Then, given that mathematicians DEFINE

i

so that

i^{2} = - 1

, simplify

- 9

2Task 2

Simplify each expression. Write your answers in terms of

i

.
(a)

- 25

(b)

- 7

(c)

- 50

3Task 3

Compute the first 8 powers of

i

i^{1}, i^{2}, i^{3}, i^{4}, i^{5}, i^{6}, i^{7}, i^{8}

. What pattern do you notice? Use the pattern to find

i^{23}

and

i^{100}

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