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# Complex Arithmetic and Vectors

Lesson 7 of 16

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

*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.

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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.

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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.

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Quadratic Equations, Day 1 of 2
- LESSON 2: Quadratic Equations, Day 2 of 2
- LESSON 3: Inconceivable! The Origins of Imaginary Numbers
- LESSON 4: Complex Solutions to Quadratic Equations
- LESSON 5: Complex Addition
- LESSON 6: The Parallelogram Rule
- LESSON 7: Complex Arithmetic and Vectors
- LESSON 8: Multiplying Complex Numbers, Day 1 of 4
- LESSON 9: Multiplying Complex Numbers, Day 2 of 4
- LESSON 10: Muliplying Complex Numbers, Day 3 of 4
- LESSON 11: Multiplying Complex Numbers, Day 4 of 4
- LESSON 12: Practice & Review
- LESSON 13: Dividing Complex Numbers
- LESSON 14: Quadratic Functions Revisited, Day 1
- LESSON 15: Quadratic Functions Revisited, Day 2
- LESSON 16: Complex Numbers Test