Complex Arithmetic and Vectors

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SWBAT interpret complex numbers as vectors in a variety of contexts. SWBAT use addition and subtraction of complex numbers to solve problems involving vectors.

Big Idea

Recognizing an application for complex numbers - vector addition - helps to give them meaning.

Reviewing Homework & Setting the Stage

15 minutes

N.B.  This lesson addresses a content standard typically reserved for Year 4 courses, and it assumes students are already familiar with vectors from their science courses.

I'll begin class by asking students to take out their homework from the previous night and to begin comparing their solutions as I take attendance.  If they don't have the same solution, then they ought to begin a conversation to see whose is correct.  Typically this allows students to clear up some difficulties, but if there are any left, we can review two or three of the problems as a class.  When we do, I like to ask for students to volunteer to explain the solutions at the whiteboard.  I've found that this is great for the students who are doing the explaining, and gives the students who are listening a little more freedom to challenge the explanation until it makes sense. (MP 3 & 6)

By the time we're done with this, I'll make sure that the parallelogram rule is illustrated somewhere on the board.  Using this diagram as a reference, I'll ask, "Are any of you familiar with something like this from your science courses, especially physics?"  Since most 11th graders have had some physics, I expect that almost everyone will be somewhat familiar with vectors and vector addition.  If no one recalls this, however, I might add the vector arrows to the diagram to make my point more explicitly and then ask the question again. 

Typically, this is more than enough for the class to recall the concept, and now we're ready to move on.

Complex Numbers & Vector Addition

15 minutes

Now it's time to make the connection between vectors and complex numbers explicit.

  • How a complex number can represent a unique vector.
  • Propose a simple "physics" problem.  A swimmer trying to cross a river.
  • Have students go to the board to solve the problem.  Teacher fade into the background as much as possible.

Working with Vectors and Complex Numbers

15 minutes

Handout Complex Addition and have students work individually at first.  Circulate to observe progress and offer minimal guidance.  Move to group time for the last 5 - 10 minutes.  The rest of the problems are homework.