Multiplying Complex Numbers, Day 1 of 4
Lesson 8 of 16
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.
Powers of i
I will begin this lesson with a brief look at the powers of the imaginary unit, i. These have already come up once or twice in class, but before we begin multiplying complex numbers, it's important to activate this prior understanding. Beginning with i and i^2, we'll work quickly through the next few powers of i. Along the way, we will constantly reiterate the fact that i^2 = -1, and we'll use this fact to simplify one power of i after another. Before long, the class should have a pretty long list of powers and should clearly see the repeating pattern of i, -1, -i, 1. (MP 8)
Once this pattern is clear, it's time to dive into multiplication of complex numbers.
Now, let the class know that today they're going to be considering multiplication with complex numbers. Just as with addition, there is an arithmetic aspect to this operation and there is a geometric aspect. (Do they recall both the arithmetic and geometric aspects of addition?)
In terms of arithmetic, multiplying complex numbers is pretty straightforward. Use the distributive law, remember that i^2 = -1, and then combine like terms. But what about the geometric interpretation? Will this create a parallelogram like addition does? The problem set Multiplying Complex Numbers will guide students to the answer.
Before you hand out the problems, however, I'd explain the definitions given on the first page. The relationship between "distance" and "absolute value" should be familiar from the real number line, but the "argument" will be something new. For now, students will just have to trust you that these two quantities will help them to think about the geometric interpretation of multiplication.
Now, pass out the problem set and break the class into small groups. Continually move back and forth between group work and whole-class discussion. As groups begin to finish problem 1, initiate a summary discussion of their solutions. The point is that multiplying by i is equivalent to a rotation of 90 degrees. For the summary discussion of #2, the point is that multiplying by a real number is equivalent to scaling the distance. Finally, for the summary of #3, the point is that multiplying by a purely imaginary number is equivalent to scaling and rotating by 90 degrees. (During this first lesson, I would not expect to get much beyond #1.)
Alternatively, you might consider running taking a Socratic approach to this lesson.
The end of class will depend on how far we have progressed today. A 3-2-1 Exit Ticket is an appropriate formative assessment on a day like this one, since it gives students a chance to reflect on the concepts we've covered, and not just the skills . I expect that many students will still be a little mystified by the geometric interpretation (MP2), and perhaps by the powers of i, but I'll have to wait for the exit tickets to know for sure.