Download Deborah A - Deborah Proulx (formerly Deborah Frank)

Lesson Plan Title: Imaginary Numbers
Grade: 11th
Subject: Algebra 2 Trigonometry
Duration of Lesson: 86 minutes
Lesson Summary/Overview:
Today’s lesson will introduce the concept of imaginary numbers. Students will learn the
definition of imaginary numbers, the value of i, how to express the square root of
negative numbers using i, how to determine the value of i to any power including powers
greater than 3, and how to solve quadratic equations with real coefficients that have
complex solutions.
Common Core Learning Standards:
N.CN.1 Know there is a complex number i such that i2 = –1, and every complex
number has the form a + bi with a and b real.
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions
Objective:
 Given practice problems involving imaginary numbers, students will demonstrate
the ability to correctly and completely express the square root of a negative
number as the product of a real number multiplied by the imaginary unit i.
 Given examples of i to powers greater than 3, students will demonstrate the ability
to correctly and completely represent the imaginary number as 1, i, -1, or –i.
 Given quadratic equations with real coefficients with complex solutions, students
will demonstrate the ability to correctly and completely solve for the roots and
express them in a + bi form.
Scope and Sequence:
Students must have prerequisite knowledge of division, reducing radicals, and solving
quadratic equations (using completing the square and the quadratic formula) with real
solutions. Following this lesson, students will be able to do the same with complex
numbers.
Materials:
 Notes 2.4/Practice 2.4
 Calculators
Routines (5 minutes):
As students enter:
 Have Power Point on Screen detailing what to pick up on front table, Today’s
HW Assignment (Practice 2.4 evens), Upcoming Tests and Quizzes
 Take attendance and pass out calculators
 Have Students clear desks except last night’s homework, pencil and calculator
Quiz/Check Homework (20 minutes):
 Give out quiz – Factoring and Radicals

Check/grade Homework – p. 196 (3,6,9,12,13,15,19-23)
Answer Questions on Prior Night’s Homework (15 minutes)
The Lesson:
Imaginary Numbers (15 minutes):
 Have Students turn to Notes 2.4
 Explain Imaginary Numbers
o Up until now you’ve been told that you can’t have a negative under the
radical – can but not in real number system
o Can be written as a real number multiplied by the imaginary unit i, which
is defined by the property i2 = -1
o Imaginary numbers have a variety of essential, concrete applications in
science and engineering
 Solve x2 + 1 = 0 on Elmo, explain solution, check
o Explain i to powers 0 through 3
o Explain definition of square root of negative k when k>0. Express as i
times square root of k
o Have students pair up
 Give students 4 problems from Notes 2.4 involving a negative
square root
 Have them work together to rewrite using real components and i
o Bring group back together and discuss answers and questions
Exponents of i greater than three (10 minutes):
 Explain process of solving by dividing exponent by 4
o Remind students that since i2 = -1, i4 = 1 (every 4 i’s is 1)
o Have students pair up
 Give students 4 problems from Notes 2.4 involving i to powers
higher than 3
 Have them work together to determine solution, 1, i, -1, or –i.
o Bring group back together and discuss answers and questions
Solving Quadratics with real coefficients and complex solutions (15 minutes):
o Explain – can use same processes we used with problems that have real
solutions to solve problems with complex solutions.
o Have students pair up
 Give students quadratic equations from Notes 2.4
 Have them solve using completing the square and/or the quadratic
formula
o Bring group back together and discuss answers and questions
Closing (5 minutes):
 Check for any remaining questions
 Show of hands – Got It All, Got Most of It, Totally Confused
 Collect Calculators
Equity/Accessibility:
 Each student can work at own pace – work completed will be dependent on time,
not amount of work
 Problems of varying difficulty
 Students can get assistance from teachers and peers
 Ask and answer questions throughout the lesson
Assessment:
 During pair work “tour” room to assess understanding – ask clarifying and
guiding questions
 Quiz
 Grade Homework
References:
Gantert, A. X. (2009). Algebra 2 trigonometry. New York, NY: AMSCO School
Publications, Inc.
Higgins, Julie. Cooperating Teacher. Guilderland High School. Albany, NY.