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# Arithmetic with Complex Numbers

Lesson 6 of 15

## Objective: SWBAT perform arithmetic with imaginary numbers and understand that imaginary numbers are useful for representing two dimensional quantities like vectors.

## Big Idea: Imaginary numbers are used to represent quantities that have two parts; working with these numbers is similar to working with polynomial expressions.

*90 minutes*

#### Warm-up

*15 min*

While I visit each table checking students' homework with the homework rubric, students work on Warm Up Complex Numbers. This file is a collection of four problems that students complete on the TI NSpire calculator [MP5]. The first two problems are a review of polynomial operations and the second two require students to solve quadratic equations in the complex number system.

I like to present these questions through the Nspire Navigator system so that I can quickly see if students have the background knowledge required to understand operations with imaginary numbers. If the warm up work indicates that students need review of polynomial operations I will spend some time with this before proceeding with the lesson.

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Based on the narrative at Cut-the-Knot, I provide my students with a brief history of the number system we currently use. I draw a diagram on the board like complex numbers venn diagram to support the discussion. In presenting the story, I emphasize that the names given to the various sets of numbers indicate the skepticism that surrounded their usefulness. I think this helps my students overcome their own skepticism about the usefulness of imaginary numbers.

My students do not have enough mathematical background yet to truly understand the usefulness of complex numbers, but I try to give them some intuition that complex numbers are useful in representing quantities that have two parts. For example, in a simple computer program, adding a complex number might move an object both to the right and up at the same time. Likewise I tell my students that because operations with complex numbers have uses in computer animation, wave theory, quantum mechanics, electricity, and other fields, we learn the basics of these operations in Algebra 2 [MP4].

After this introduction, I provide direct instruction in the following:

- simplifying powers of i
- standard form of a complex number
- complex conjugates
- adding and subtracting complex numbers
- multiplying complex numbers
- rationalizing the denominator of a complex number

#### Resources

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To practice operations on complex numbers, students complete Complex Number Maze Activity. This activity asks students to simplify expressions with complex numbers and then find a path through the results that are non-real. This draws attention to the concept that performing an operation on two complex numbers can produce a real or non-real result.

As my students work, I use the 3-Cup System to determine who needs my help [MP1].

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#### Exit Ticket and Assignment

*15 min*

To make sure that students understand basic operations on complex numbers, I use the last 10 minutes to administer a short formative assessment via the TI Navigator system, Complex Number Exit Ticket. Administering this short assessment on the calculators allows me to get a quick read on how well my students understand the day's lesson [MP5].

For homework, students will work through the 3 page worksheet Complex Numbers Intro. In this exercise set, students will review some of the notes from today's class and practice some of the skills they learned. The solutions to this assignment will be available on Edmodo.

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- LESSON 1: Seeing Structure in Expressions - Factoring Higher Order Polynomials
- LESSON 2: Proving Polynomial Identities
- LESSON 3: Polynomial Long Division and Solving Polynomial Equations
- LESSON 4: The Remainder Theorem
- LESSON 5: The Fundamental Theorem of Algebra and Imaginary Solutions
- LESSON 6: Arithmetic with Complex Numbers
- LESSON 7: Review of Polynomial Roots and Complex Numbers
- LESSON 8: Quiz and Intro to Graphs of Polynomials
- LESSON 9: Graphing Polynomials - End Behavior
- LESSON 10: Graphing Polynomials - Roots and the Fundamental Theorem of Algebra
- LESSON 11: Analyzing Polynomial Functions
- LESSON 12: Quiz on Graphing Polynomials and Intro to Modeling with Polynomials
- LESSON 13: Performance Task - Representing Polynomials
- LESSON 14: Review of Polynomial Theorems and Graphs
- LESSON 15: Unit Assessment: Polynomial Theorems and Graphs