Like our work with vectors – today’s lesson is going to have important implications as we start to look at polar coordinates later in the semester. One of the big ideas for the day is that we can start at the origin and end at another by moving right or left and up or down – or we can rotate to a certain angle and move out radially from the origin. Another important focus of the day is that we can look at complex numbers through a graphical lens. Students will probably only have used the algebraic perspective of complex numbers, so it will be an intellectual challenge for them to see these numbers in a different way.
As class begins, I will go over the front of this worksheet together with my students. A common misconception my students embrace is the idea that complex numbers have to include an imaginary part, so I make sure that they understand the definition. I find that showing a Venn diagram with imaginary and real numbers as subsets of the set of complex numbers may help to erase this confusion.
In terms of the tasks on the worksheet, my students will usually correctly guess the associated ordered pairs with the complex number, but Question # 3 will probably require some direct instruction. Drawing a diagram is key to opening up students to understand the trigonometric form of complex numbers. I always make sure that students see the right triangles when graphing these points. I also make a big deal about how the process of writing the trig form of a complex number: it is just like our work with vectors. We are using the horizontal and vertical components to find the magnitude and the direction. This idea will come up again when we study polar coordinates in a later unit.
Question #4 can be started as a class, but I plan to give students some time to work on the calculations themselves. As they work I will look to see if students can recognize that putting them both in trig form or both in a + bi form would simplify the problem immensely. Since one of the goals is to see what happens in trig form, encourage them to try that approach. I give students about 10 minutes to work on part a) and b) for this problem.
When students work on Question_#4, there are two issues I usually see in students' attempts at the problem. One is that they may get bogged down by logistics of multiplying 5(cos60° + isin60°) by 2(cos135° + isin135°). So, when I choose a student to share, I make sure that their work is clearly organized and easy to follow. Another complication is that students may not be able to simplify if they do not remember the cos(A + B) and sin(A+ B) formulas. Leaving things unsimplified motivates the conclusion that we simply multiply the r values and add the angles measures together when multiplying complex numbers together.
Teacher's Note: Getting the correct r value when converting from a + bi form to trig form can be difficult for some students. In this video I discuss some ways to approach this.
After a student shares, I always give one more multiplication problem to see if they can apply the shortcut. Multiplying complex numbers is tedious – so I always give more examples until they are comfortable with the process. Finally, I ask students to generalize the multiplication rule using variables in Question #5. I let them begin by discussing the task at their tables.
As a concluding challenge, I ask students: "Can guess what the division rule for the trig form of complex numbers is?" My students usually guess right away after our work with multiplication.