This lesson aims to teach students to represent addition of complex numbers, including complex conjugates, in the complex plane. This is a "plus" standard (HSN.CN.B.5) usually included in Precalculus, but since complex numbers are difficult to conceive abstractly, I find it very helpful for students to connect them to a geometric representation. The parallelogram rule for the addition of complex numbers may resonate with many students because of their familiarity with vector arithmetic from physics courses, and it will certainly bring back memories from geometry for everyone.
At the beginning of class, we will quickly review homework and then use those solutions as a springboard for beginning our investigation of addition in the complex plane. I'll begin by asking students to put their solutions to last night's homework on the board (see previous lesson). Once this is done, we'll have five different pairs on complex conjugates plotted in the same complex plane. Once any questions regarding the homework have been cleared up, I'll select one complex conjugate pair and ask the students what we'll get if we add them together. After confirming the answer with the class (see the whiteboard), I'll give them two minutes to quickly add together the remaining pairs of complex conjugates.
What will they notice? The sums are all real numbers! This leads me to ask, "Is the sum of two complex numbers always a real number?" To answer this question (and lead to the discovery of the parallelogram rule) I'll select five pairs of complex numbers from those on the board. In the next section of the lesson, students will work individually to add those five pairs of numbers and to plot the addends and sum in the complex plane.
For further details, see the video resource.
Students will work individually on the assigned addition problems (five pairs of non-conjugates chosen more or less randomly from solutions to last night's homework). I don't expect these to be a big challenge, but I think it's important for the students all to have the chance to "discover" for themselves the parallelogram rule for complex addition. (MP 8) I will circulate through the class to offer guidance to any students who might need it. In particular, I want to make sure that no one is trying to combine imaginary and real terms and that everyone is plotting the numbers correctly on the plane. For more information, please see my strategy video on Individual Time.
My aim with this little exercise is for the class to recognize the geometric interpretation of complex addition. I might offer the following hints, if they seem necessary:
"If you were given just the two points in the complex plane, without knowing their value, could you make a reasonable guess at their sum?" To drive this point home, I might put two points on the plane, but not at grid points. They will have to think about the parallelogram to find the sum.
"Since these are points in a plane, you might consider the geometry of the situation! The three points make a triangle, but if you include (0, 0) what shape do they make?"
Once students have completed the addition problems and have begun to recognize deeper connections, it's time to move into the next section of the lesson.
After 10 minutes, it's time to summarize what the students have found. I'll begin by asking whether anyone noticed any patterns as they worked out the solutions to these problems. You never know what you'll get with a question like that, but I'm looking for someone to mention the parallelogram shape. When someone does, I'll ask that student to come to the board to show us exactly what they're trying to describe. (MP 3)
Beginning with this example, I'll ask the rest of the class to confirm this pattern or rule. When everyone sees that it holds for the five problems they did, I'll ask, "Why? Why should it be a parallelogram?" I expect students to offer a variety of justifications, but I will make sure that we consider the slopes of the line segments joining the points in the plane. If it turns out that no one is quite sure what to say, I'll use a pair-share strategy to see if they can generate some ideas in that way. Before long, we should be able to convince ourselves that it is reasonable for addition of complex numbers to be associated with parallel lines and thus a parallelogram. (See this link for a great dynamic tool.)
While everyone is feeling satisfied with this new understanding, I'll change the subject (seemingly) by asking about subtraction. After a moment's reflection, I expect students will begin to see that the parallelogram rule will still hold. An example problem will help me to determine whether the class has really grasped the concept, and I'll make a point of cold-calling on the students who are typically quiet (or the ones who haven't contributed to the conversation thus far) to describe the steps of the solution. Once I'm satisfied that the relationship between the numeric solution and the graphical solution is understood, it's time to move on.
With about 10 minutes left in class (hopefully more if the students were quick to pick up on the vector connection!), I'll assign the the problems on the Complex Addition handout. These are simple addition and subtraction problems with complex numbers, as well as some quadratic equations having complex solutions.
Tonight's homework is to complete the handout and make progress on the Weekly Workout.