Multiplying Complex Numbers, Day 2 of 4

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SWBAT use the unit imaginary number and the field axioms to multiply complex numbers. SWBAT represent and interpret multiplication of complex numbers in the complex number plane.

Big Idea

Students explore and explain correspondences between numerical and graphical representations of arithmetic with complex numbers.

Looking Back

5 minutes

"So, where were we?"

I'll begin class with this question, which should prompt responses from many students.  Using these responses as a springboard for more questions and asking students to clarify or to sketch a picture at the board, we will quickly review and summarize our investigation from the previous lesson.

During this discussion, I'll be sure to bring up some of the questions students wrote down on their exit tickets at the end of that lesson.  It's important to clear up as many of those as we can before we move on.

Multiplication of Complex Numbers

30 minutes

During this lesson, I want to help students come to several conclusions.

  • First, when a complex number is multiplied by i, it's distance from 0 is unchanged, but its argument increases by 90 degrees.
  • Second, when a complex number is multiplied by a real number, it's distance from 0 is multiplied by that factor, but its argument is unchanged.
  • Third, when a complex number is multiplied by an imaginary number (that is, by both a real number and i), then it's affected in both ways.  It's distance is multiplied and its argument is increased by 90 degrees.
  • Finally, we come to the question of the product of two complex numbers.  The students may guess that both the argument and distance will be affected, but the final goal is to help students articulate the fact that the distances of the two numbers are multiplied while the arguments are added. (MP 8)

As in the previous lesson, I will do this by asking students to work in small groups on one problem at a time from Multiplying Complex Numbers.  We will pick up wherever we left off yesterday, and the group-work will be frequently interrupted for summary class discussions.  As much as possible, I will rely on students to provide the necessary explanations. (MP 2) (See this video for some details.)

Wrapping Up

5 minutes

The class conversation over the last two lessons has been pretty heady, and many students may need some time to digest it all.

Tonight's homework will help them to do just that by taking them back to the basic concepts of the distance and argument of a complex number.  Students should complete Practice 1 tonight.  They should plot the given number in the plane, use the Pythagorean theorem to find the distance/absolute value, and then use trigonometry to find the argument.  Once they understand what is expected, most of the class should find this refreshingly simple.