Vector Walks Slides Vector Walks
Mapping Complex Numbers to Geometry
Complex Plane
Vector Addition
Warm-up
5 MINUTE CHALLENGE
Draw these vectors from the origin on your grid paper:
1 \(\vec{u} = \langle 3, 2 \rangle\)
2 \(\vec{v} = \langle -4, 1 \rangle\)
"Remember: A vector is a directed line segment. Start at (0,0) and move to the target coordinates!"
Use your grid paper now
Visualizing the Plane
Watch the plotting method carefully
Embedded media
Key Question:
How does plotting \(3+2i\) compare to plotting the vector \(\langle 3, 2 \rangle\)?
The Anatomy of a Complex Vector
Re
The Real Part (\(a\))
Horizontal movement on the x-axis.
Im
The Imaginary Part (\(b\))
Vertical movement on the y-axis.
\(a + bi \longleftrightarrow \langle a, b \rangle\)
Re Im 3 + 3i
Main Activity
The "Vector Walk" Challenge
1
Start Strong
Plot the first complex number as a vector starting from the origin (0,0).
2
Keep Walking
Plot the second number, but start your vector from the head (tip) of the first one!
3
Final Destination
Draw the resultant vector from (0,0) to where you finished. What number is that?
Use your colored pencils for each vector!
Reflection
"How does our 'Vector Walk' relate to combining like terms algebraically?"
\((3 + 2i) + (1 - 4i)\)
Horizontal: \(3 + 1 = 4\)
Vertical: \(2i - 4i = -2i\)
Vector Walks Worksheet Vector Walks Worksheet
Algebraic Form to Geometric Vectors
Name:
Date:
Part 1: Vector Review
Draw the following vectors from the origin on the small grids below.
1. \(\vec{u} = \langle 3, 2 \rangle\)
2. \(\vec{v} = \langle -4, 1 \rangle\)
Part 2: Video Check-in
From the video: Plot the following complex numbers as vectors.
a) \(3 - 2i\)
Movement: ______ right, ______ down
b) \(2i - 4\)
Rewrite in \(a+bi\) form first: __________
Part 3: The Vector Walks
Use colored pencils: Vector 1 (Blue) , Vector 2 (Red) , Resultant (Green) .
Challenge 1
\((2 + 3i) + (4 - 1i)\)
Step-by-Step Analysis:
1. Walk 1: Start at (0,0), move to __________.
2. Walk 2: Start from last stop, move __________.
3. Resultant: What is your final coordinate?
Re Im
Challenge 2
\((-3 + 2i) + (1 + 2i)\)
Step-by-Step Analysis:
1. Walk 1: Start at (0,0), move to __________.
2. Walk 2: Start from last stop, move __________.
3. Resultant: What is your final coordinate?
Challenge 3
\(4i + (2 - 5i)\)
Step-by-Step Analysis:
1. Walk 1: Start at (0,0), move to __________.
2. Walk 2: Start from last stop, move __________.
3. Resultant: What is your final coordinate?
Complex Plane Grid Paper Complex Plane Grid
Real (Re)
Imaginary (Im)
Real Axis (Re)
Imaginary Axis (Im)
0
5
10
15
-5
-10
-15
5i
10i
15i
-5i
-10i
-15i
Use colored pencils to distinguish between multiple vectors on this plane.
Pre-Calculus: Geometry of Complex Numbers
Vector Walks Facilitator Guide Facilitator Guide
Vector Walks Lesson Plan
Time
40m
Objective
Students will demonstrate the connection between the algebraic sum of complex numbers and the geometric "head-to-tail" addition of vectors on the complex plane.
Key Vocabulary
• Complex Plane
• Real Part (a)
• Imaginary Part (b)
• Resultant Vector
• Head-to-Tail Method
Instructional Sequence
5
min
Warm-up: Vector Recalibration
Students draw basic vectors \(\langle 3, 2 \rangle\) and \(\langle -4, 1 \rangle\) on small grids. This activates prior knowledge of the coordinate plane and directed line segments.
10
min
Video: The Complex Plane Anatomy
Watch the video. Pause at 2:55 to ask: "What are \(a\) and \(b\) in \(2i - 4\)?". Address the order misconception. Use the slide's key question: "How does \(3+2i\) compare to \(\langle 3, 2 \rangle\)?"
20
min
Activity: Vector Walks
Pairs work through the worksheet. Ensure students use colored pencils. One student "walks" the first vector, the other "walks" the second starting from where the first ended.
Tip: Circulate and check Challenge 3 (\(4i + (2-5i)\)). Students often struggle with pure imaginary numbers (0 horizontal movement).
Answer Key
Part 2: Video Check
a) \(3 - 2i\): 3 right, 2 down
b) \(2i - 4\): \(-4 + 2i\)
Part 3: Vector Walks
1. \((2+3i) + (4-1i) = \mathbf{6+2i}\)
2. \((-3+2i) + (1+2i) = \mathbf{-2+4i}\)
3. \(4i + (2-5i) = \mathbf{2-1i}\)
Common Misconceptions
"Real numbers aren't complex": Randy addresses this at 5:17. Remind students that \(4 = 4+0i\). It still has a "vector" form (horizontal only).Head-to-Head: Some students will draw both vectors from (0,0). Emphasize that addition is a *sequence* of movements. "Wait for the first person to finish before you start your walk."$i$ as a variable: Remind students that $i$ is not a variable to be solved, but a unit defining a specific axis. 
