Muliplying Complex Numbers, Day 3 of 4

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Objective

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

Practice helps students uncover hidden patterns and discover simple rules.

Reviewing Homework

10 minutes

In the previous lesson, students learned the definitions of the absolute value (i.e. distance or modulus) and argument of a complex number, and they were assigned Practice 1 for homework.  I like to begin class today by reviewing the answers to this assignment. To do this, I like to ask students for the answers to the problems.  Rather than say "right" or "wrong", I'll ask the class if they agree.  If they do agree (and they're all correct), great.  If not, then we'll clear up the difference as quickly as we can.

Invariably, a number of students will have some trouble when the argument of a complex number is greater than 90 degrees, so I'll plan on discussing these cases with the class.  See the video for details.

Multiplication Practice

25 minutes

Setting the Stage:

Now, I assign Practice 2 for classwork.  As I hand it out, I will point out to the students that they now have two complex numbers in each problem.  Just as before, they are to plot the numbers in the plane and then compute both the distance and argument for each.  Next, they are to multiply the two numbers to obtain the third complex number, zq.  Again, they should plot this number and compute its distance and argument.  Any questions?  There will probably be questions, so it might be a good idea to get started on the first one together.

Practice!

For this lesson, the students are free to work individually or in small groups, as long as they are on task.  They will be checking their answers against one another, and I will be circulating among them offering encouragement and answering questions.  The students should quickly notice that they're give the same numbers for z that they had on Practice 1 (I was feeling generous), so they can copy over their solutions from that sheet.

As they complete these problems, many students will notice the pattern that the distances are being multiplied while the arguments are being added.  Excellent!  I encourage them to look for these patterns but also to remain somewhat skeptical when they find them. (MP 8)  Keep plugging away with an eye out for counter examples.

Describing Patterns

10 minutes

Through the completion of these practice problems, my less proficient students will have grown more confident with multiplication of complex numbers.  They will be feeling more comfortable plotting the numbers and finding distances & arguments.  My stronger students will have started making connections and recognizing patterns.  Along the way, they'll also be gaining valuable practice with right-triangle geometry and trigonometry.

During these final ten minutes, I want to give the class the chance to share the patterns they've discovered with their peers.  I'm hoping they'll have seen the following:

    1. Multiplying by i rotates the number 90 degrees.
    2. Multiplying by a real number (other than 1) changes the distance.
    3. The distances of the two factors are multiplied to get the distance of the product.
    4. The arguments of the two factors are added to get the argument of the product.

As class ends, I will hand out Practice 3 and ask everyone to complete the first problem tonight for homework.  (Why so much practice?  See this video.)