The Parallelogram Rule

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SWBAT to show that addition of complex numbers may be represented by a parallelogram in the complex number plane.

Big Idea

A graphical context makes arithmetic with complex numbers more meaningful and concrete.

A Particular Example

10 minutes

This lesson is intended to provide a more rigorous basis for the parallelogram rule students identified in Complex Arithmetic.  My focus here is on student discourse & questioning, rigor, and conceptual understanding. I will have students at the board as much as possible, I will encourage them to ask questions, and I will reiterate again and again that the goal is to understand. Please see the video for a general overview.

I begin class with an addition problem on the whiteboard, (2 - 3i) + (-5 + 2i), and a complex number plane next to it.  Then I ask for a volunteer to demonstrate to the class how to add the two numbers and how to represent the addition in the complex number plane.  This is a fairly simple demonstration, so it is an opportunity to look for a volunteer who doesn't usually put him/herself forward.  During the demonstration, I try to fade into the background and I direct any class questions to the student at the board for explanation.  Once the addition is complete and the numbers have been placed in the complex number plane, I'll ask this student to wrap things up by drawing for us the parallelogram formed by these three points and the origin. 

Next, I ask for a second volunteer to explain to the class how they know that the quadrilateral formed by the two addends, the sum, and the origin must be a parallelogram.  I'm expecting the argument to focus on the slopes of the line segments based on the known values of the numbers.

An Intermediate Example

15 minutes

Now, leaving the previous problem & its explanation on one half of the board, I use the other half to set up a new problem: (5 - 7i) + (a + bi).  Again, the task is to add the two numbers, put the addends and the sum on the complex number plane, and then prove that they form three vertices of a parallelogram.

The students will work completely independently for about 5 minutes, while I circulate to see what kind of progress they're making.  Next, I'll direct them to begin working together, and I'll make some suggestions about who should work with whom. (See my reflection on this below.) Along the way, I'll step in frequently with hints, suggestions, and partial explanations to keep things moving.  Finally, one or two students will be asked to go the board to explain the whole solution to the class.  I will try to focus the classes attention on the following:

  • (a + bi) stands for any complex number
  • slopes of the line segments should be given in terms of a and b
  • the solution method/process is identical to the one we used in the previous section of the lesson

I will need to explain to many students that while (a + bi) is supposed to be arbitrary, they'll have to pick a place to put in on the plane.  This may need some clarification, so I'll tap into their knowledge of geometry with an analogy.  In that class, we use a single, generic triangle to prove things about all triangles; in this class, we're using one generic number, (a + bi), to stand for all complex numbers.

A Totally General Proof

20 minutes

Finally, I will pose the completely general question: If we add the numbers (a + bi) and (c + di) and then represent the addition in the complex number plane, will they form the vertices of a parallelogram?

Unless I have a really advanced class, I'll probably need to guide the students through this proof Socratically.  Staying at the whiteboard, I like to point out that this is no different from the problems we've just done except that the particular numbers have been replaced with variables.  First, we'll select positions for (a + bi) and (c + di) more or less randomly in the plane.  Then we'll discuss what the sum must be, (a + c) + (b + d)i, and where it should be placed in the plane.

This brings us immediately to the notion of slope because to put the sum on the plane, we'll start at (a + bi) and add c, moving c units horizontally. Then we'll add di, moving d units vertically.  Not only does this movement bring us to the point representing the sum, but it also reveals the slope of the line segment, d/c.  From here, we'll determine the slopes of all four segments and observe that the figure must be a parallelogram.

There you have it!  We've just proven that complex addition can always be represented with a parallelogram in the complex plane!  This is pretty neat when you consider that even addition of real numbers fits this pattern - but the parallelogram degenerates into a straight line!

The students have done a lot of hard thinking today, so I'll congratulate them on what they've accomplished and give them a night without homework.