
Teacher 
Milton Francis 
Summary 
In this lesson, students will reinforce their previous knowledge of complex numbers by performing arithmetic operations on them. That is, they will add, subtract, multiply and divide using these imaginary/complex numbers. Their solution to these complex number problems will be written in the form of the sum or difference of a real number and an imaginary number, i.e. a + bi, where 'a' represents the real number and 'bi', the imaginary portion. 
Grade/Level 
1112 
Time Frame 
Two 48minute class (i.e. 2 days) 
Subject 
Mathematics 
Topic/Aim 
How do we simplify Complex Numbers, leaving the answers in the form a + bi. 
Notes 
Even though this is a NY Regents Math B lesson, it could be modified to a Math A level. In teaching it as a Math A lesson, the teacher would not conjugate. That is, he/she would not introduce the binomial a + bi as the denominator. Instead, only the monomial bi, would be used in the denominators. By using this imaginary monomial, there would be no need to conjugate, as conjugation only works with binomial denominators. This modification, however, would cause the topic of the lesson to read thus: Simplifying Complex Numbers.

Standards and Key Concepts 


Standards 
NY New York State Standards 

Subject: Mathematics, Science, and Technology (1996, 2005 Math update)


Learning Standard 3: Mathematics (2005 update) Students will: •understand the concepts of and become proficient with the skills of mathematics; •communicate and reason mathematically; •become problem solvers by using appropriate tools and strategies; through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.


Grade/Subject : Algebra 2 and Trigonometry


Area : Content Strands


Strand : Number Sense and Operations Strand


Standard : Students will understand meanings of operations and procedures, and how they relate to one another.


Performance Indicator : A2.N.9 Perform arithmetic operations on complex numbers and write the answer in the form bi a + . Note: This includes simplifying expressions with complex denominators.



Performance Indicator : A2.N.8 Determine the conjugate of a complex number



Performance Indicator : A2.N.7 Simplify powers of i












Understandings 
At the culmination of this lesson (Day 2), the students should be able to: 1. Recall the use of the imaginary number, i. 2. Determine the conjugate of a complex number. 3. Apply the four basic rules of arithmetic (addition, subtraction, multiplication, and division) in simplifying complex numbers. 4. Recognized that conjugation is only applied to the rule of division in complex numbers, and not to the other three rules. 
Essential Questions 
In a previous lesson on imaginary numbers, we rationalized the denominator of the fraction 5/2i by multiplying the fraction by 1 in the form of 2i/2i. How can we rationalize the denominator of the following fractions: 1. 6/(2 + i)and 2. (5 + 4i)/(7  3i), being mindful of the fact that in rationalizing, we multiply the fraction by 1 in a certain form, and that an imaginary number cannot be part of a resulting denominator? 
Knowledge and Skills 
To fully comprehend the lesson, students would recall the knowledge applying the four basic arithmetic rules to algebraic expressions involving binomials. For example, the could use a few minutes prior to the lesson by reviewing these links:
Links
1. Adding Polynomials Review of addition of
Polynomials
2. Multiplying Binomials Multiplication of
Binomials.
3. Subtracting Polynomials Subtracting
polynomials. 
Performance Tasks and Assessment



Performance Task 
This lesson is technologyrelated, so the students will use the Internet throughout. After reviewing the polynomial links, they will be directed to a link on the topic. This direction will serve to assist them in following instructions and the steps used in simplifying the complex numbers, while simultaneously discovering the wealth of information on the Internet with regards to the topic being learned. The link is as follows:
Links
1. Operations on Complex Numbers Addition,
subtraction, multiplication and division of
complex numbers. 
Performance Prompt 
The following interactive exercise will be done by the students as they work in pairs: Links
1. Practice Exercise on Simplifying Complex Numbers Addition, subtraction, and multiplication of complex numbers.
2. Practice Multiplying and Dividing Complex Numbers Multiplication and division of complex numbers. 
Assessment/Rubrics 
Rubrics: Complex Numbers 
Learning Experiences and Resources



Sequence of Activities 
To comprehend the lesson by the culminating exercise, the students would have done the following:
1. Differentiate between real and imaginary numbers.
2. Understand the concept of the imaginary number i.
3. Be able to simplify/not simplify like/unlike algebraic terms respectively through addition and subtraction.
4. Able to multiply binomial algebraic expressions using the FirstOuter InnerLast (FOIL) method.

Differentiated Instruction 
For my English as a Second Language (ESL) students, certain terms of the lesson like conjugate, raionalize, imaginary, etc would be written as the equivalent to the Spanish terms and be placed on the chalkboard. However, all four groups, ESL, AtRisk,Special Ed.and Advanced Learners would be working as a small group. Each group will have at least one advanced learner assigned to it, whose task is to assist the other learners to comprehend the lesson as the teacher teaches. 
Resources 
 Materials and resources:
Notebooks Pens/Pencils Printed copy of homework sheet on the day's lesson
 Technology resources:
Internet Explorer
 The number of computers required is 1 per student.
 Students Familiarity with Software Tool:
Students are familiar with the use of the Internet.
 Computer use (Internet accessibilty)
Use of notebooks and pens/pencils Distribution of printed copy of homework sheet. 