Plane Plotters Worksheet Rotation Revolution
Lesson 1: The Geometry of i
Complex Number Voyages
Student Name:
Date:
Part 1: The 180° Flip
Imagine the number line. When we multiply a positive number by -1, what happens to its position?
Algebraic: \(x \rightarrow -x\)
Geometric: 180° Rotation
If multiplying by \(-1\) is a 180° rotation, then multiplying by \(i\) (where \(i^2 = -1\)) must be two steps that total 180°.
Hypothesis: What degree of rotation does a single multiplication by \(i\) represent?
Part 2: The Complex Plane
We define the Real Axis (horizontal) and the Imaginary Axis (vertical). Plot the following complex numbers \(z = a + bi\) as coordinates \((a, b)\).
Re Im
A \(z_1 = 3 + 2i\)
B \(z_2 = -4 + i\)
C \(z_3 = -2 - 3i\)
D \(z_4 = 5i\)
Think Geometric:
If you start at \(z = 3\) and multiply by \(i\), where do you land? Does this support your hypothesis from Part 1?
Part 3: The Power Cycle
Calculate the powers of \(i\) algebraically and then sketch the path on the grid below.
Step 1
\(i^1 = \_\_\_\)
Step 2
\(i^2 = \_\_\_\)
Step 3
\(i^3 = \_\_\_\)
Step 4
\(i^4 = \_\_\_\)
Path of \(i^n\)
Voyager's Challenge
If multiplying by \(i\) is a 90° counter-clockwise rotation, how would you describe the geometric operation of multiplying by \(-i\)?
Rotation Revolution Slides Vector Voyages
ROTATION
REVOLUTION
Discovering the geometry of the imaginary unit through the lens of transformation.
Mission 01
The Dimensional Dilemma
"If multiplying by \(-1\) flips a number 180° on the line..."
"...what mathematical operation would rotate a number 90° into a new dimension?"
Where is the middle of a flip?
Defining \(i\)
Algebraic Definition
\(i^2 = -1\)
Geometric Definition
Multiplying by \(i\) is a 90° counter-clockwise rotation about the origin.
1
\(i\)
The Complex Plane
Real Axis (Horizontal)
Imaginary Axis (Vertical)
Complex numbers are like map coordinates.
\(z = a + bi\)
Right/Left Up/Down
1 \(3 + 4i\)
2 \(-2 + 2i\)
3 \(-5 - 3i\)
Target Located
Navigation Check
"If you multiply \(4 + 2i\) by \(i\), what is the resulting number? Use your 90° rotation intuition first, then verify algebraically."
Plot it.
Rotate it.
Solve it.
Vector Translations Slides Vector Voyages
VECTOR
TRANSLATIONS
Visualizing complex addition and subtraction as movements through the complex plane.
Mission 02
Complex Numbers as Vectors
In the previous mission, we plotted points. Now, we draw vectors .
A vector represents displacement from the origin \((0,0)\) to the number \(z = a + bi\).
3 + 3i
Adding Displacement
The Parallelogram Rule
Adding two complex numbers is like performing one movement, then the other.
Tip-to-Tail Method
Parallelogram Method
z₁ + z₂
Subtracting Vectors
Subtracting \(z_2\) is the same as adding the opposite vector \(-z_2\).
\(z_1 - z_2 = z_1 + (-z_2)\)
Geometric Interpretation:
The vector pointing from the tip of \(z_2\) to the tip of \(z_1\).
z₁
z₂
z₁ - z₂
Coordinate Confirmation
Let \(z_1 = 3 + 2i\) and \(z_2 = 1 + 4i\).
If you start at the origin and follow \(z_1\), then follow \(z_2\), where do you land? Does this match the algebraic sum?
Verification Rule:
\((a + bi) + (c + di) = (a + c) + (b + d)i\)
Vector Voyage Log Worksheet Vector Voyage Log
Lesson 2: Vector Translations
Complex Number Voyages
Student Name:
Date:
Task 1: Drawing Displacements
Represent each complex number as a vector starting from the origin \((0,0)\).
A) \(z_1 = 4 + 3i\)
B) \(z_2 = -2 + 5i\)
Task 2: The Parallelogram Rule
Given \(z_1 = 2 + i\) and \(z_2 = 1 + 3i\).
Algebraic Verification
\(z_1 = 2 + 1i\)
Observation:
How do the coordinates of the resulting point compare to your algebraic sum?
Task 3: Subtraction as "Difference"
Sketch \(z_1 = 5 + 4i\) and \(z_2 = 2 + i\). Then, draw the vector representing \(z_1 - z_2\).
Voyager Reflection:
If addition is like walking two steps in sequence, how would you describe subtraction in terms of walking on the map?
Spiraling Out Slides Vector Voyages
Spiraling
Out
Unlocking the geometry of multiplication: The dance of rotation and dilation.
Mission 03
Beyond the 90° Turn
We know \(i\) rotates everything by 90°.
What if we multiply by \(1 + i\)?
Is it still just a rotation? Or does the size of the vector change too?
Rotation
The Angle (Argument)
Dilation
The Length (Modulus)
The DNA of a Vector
Modulus (\(r\))
The length from origin to point.
\(|z| = \sqrt{a^2 + b^2}\)
Argument (\(\theta\))
The angle from the positive Real axis.
\(\theta = \arctan(b/a)\)
z
Length r
θ
The Voyager's Theorem
When you multiply complex numbers...
Lengths
MULTIPLY
\(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)
Angles
ADD
\(\text{arg}(z_1 \cdot z_2) = \theta_1 + \theta_2\)
"It's a rotation and a stretch happening at the same time."
The Growth Spiral
Let \(z = 1 + i\).
What happens if we multiply \(z\) by itself over and over?
\(z^2\)
Rotates 45°, scales \(\sqrt{2}\)
\(z^3\)
Rotates another 45°...
\(z^4\)
Total 180° rotation?
Spiral Explorer Activity Spiral Explorer
Lesson 3: Multiplication Patterns
Complex Number Voyages
Student Name:
Date:
"In this mission, you will investigate how multiplication changes the length and the angle of complex numbers. You are looking for a hidden geometric rule."
Task 1: The \((1+i)\) Sequence
Calculate the algebraic value of each power of \(z = 1 + i\). Then, calculate its modulus \(|z|\) and its argument \(\theta\).
Term Algebraic (\(a+bi\)) Modulus (\(r\)) Argument (\(\theta\)) \(z^1\) \(1 + i\) \(\sqrt{2}\) 45° \(z^2\) \(z^3\) \(z^4\)
Sketch the powers here
Discovery Log
Look at your results for modulus and argument. What is happening as the exponent increases?
Task 2: The Multiplication Rule
Based on your observations, fill in the blanks for the geometric definition of complex multiplication:
1
To find the length of the product, you must ______________________ the lengths of the original numbers.
2
To find the angle of the product, you must ______________________ the angles of the original numbers.
Visual Proof Challenge
If you multiply a number by \(2i\), how does it move? (Describe the rotation and the dilation).
Voyager's Riddle
"I am a complex number. When you multiply me by myself, I end up on the negative real axis, and I am twice as far from the origin as I was before. Who am I?"
Mirror Worlds Slides Vector Voyages
Mirror
Worlds
Exploring the geometric symmetry of the complex conjugate.
Mission 04
The Mathematical Mirror
"Every complex number has a twin on the other side of the real axis."
\(z = a + bi\)
\(\bar{z} = a - bi\)
Mirror Plane (Real Axis)
z
\(\bar{z}\)
Crossing the Streams
What if we multiply?
When a number meets its mirror image, the imaginary parts cancel out.
\(z \cdot \bar{z} = a^2 + b^2\)
The Discovery:
The product is always a Real Number.
Specifically, it is the square of the distance from the origin!
\(z \cdot \bar{z} = |z|^2\)
Symmetry in Action
Conjugation doesn't just work on single numbers. It reflects whole operations.
Sum \(\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2\)
Product \(\overline{z_1 \cdot z_2} = \bar{z}_1 \cdot \bar{z}_2\)
"The geometry remains consistent, whether you operate and reflect, or reflect and operate."
Mirror's Edge
If \(z = 2 + 2i\), find \(\bar{z}\). Then, find the angle of both.
Observation Task
"How does reflecting across the Real axis change the argument (angle) of the vector?"
Reflection Lab Worksheet Reflection Lab
Lesson 4: The Complex Conjugate
Complex Number Voyages
Student Name:
Date:
Task 1: Drawing the Reflection
1. Plot \(z_1 = 3 + 4i\).
2. Draw its reflection across the Real axis. This is the conjugate, \(\bar{z}_1\).
Algebraic \(\bar{z}_1 = \_\_\_\_\_\_\_\_\)
The Rule:
The complex conjugate \(\bar{z}\) of \(a+bi\) is \(a-bi\). Geometrically, this is a reflection over the horizontal axis.
Real
Imaginary
Task 2: Crossing Mirror Boundaries
Calculate the product of \(z = 2 + 5i\) and its conjugate \(\bar{z}\). Show your steps.
Algebraic Work
Geometric Result
Where does the result lie on the complex plane?
Key Insight:
"Multiplying a complex number by its conjugate always produces a real number because..."
Task 3: Neutralizing the Imaginary
In algebra, we use the conjugate to "rationalize" denominators. Use your understanding of the "Mirror World" to simplify this expression:
\(\frac{1}{3 + i}\) [\(\text{Multiply by } \frac{3-i}{3-i}\)] \(\frac{\_\_\_\_\_\_\_\_}{\_\_\_\_\_\_\_\_}\)
Voyager's Debrief
If you add a number and its conjugate (\(z + \bar{z}\)), is the result real, imaginary, or complex? Explain using a sketch or algebra.
Complex Treasure Hunt Slides Vector Voyages
Complex
Treasure Hunt
Synthesis Mission: Using algebra and geometry to navigate the plane.
Mission 05
The Explorer's Protocol
To find the treasure, you must follow a series of "Vector Instructions".
A
Algebraic Clues
G
Geometric Clues
Your Toolkit
Addition = Translation
Multiplication = Rotation & Dilation
Conjugation = Reflection
\(i\) = 90° CCW Rotation
Sample Navigation
Clue 01:
"Start at \(1+i\). Multiply by \(3i\)."
Geometric Strategy:
Rotate your current position by 90° and stretch its distance by 3.
1+i
-3+3i
Algebra vs Geometry
When to switch gears?
Geometry is Best for...
Rotating by multiples of 90°, stretching/shrinking lengths, and reflecting across axes.
Algebra is Best for...
Precise calculations, adding complex coordinates, and verifying geometric hunches.
Final Destination
"The treasure is located at the result of your five-step journey."
GOAL
Identify the final complex coordinate \(z\) and sketch its vector path on your Voyager's Map.
Voyager's Final Map Activity The Voyager's Final Map
Lesson 5: Synthesis Mission
Vector Voyages Sequence
Student Name:
Date:
"You are at the helm of the S.S. Imaginary. Follow the sequence of commands below to navigate through the complex plane and locate the sunken treasure. For each step, decide whether an algebraic calculation or a geometric transformation is more efficient."
Voyager's Chart
Flight Log
01. Departure
Start at \(z_0 = 1 + i\).
02. The First Turn
Multiply by \(2i\). Where are you now?
03. The Drift
Add \(3 - 4i\). What are your new coordinates?
04. Mirror Pass
Reflect across the Real axis (take the conjugate).
05. Final Pulse
Multiply by \(\frac{1}{2}\). This is your destination.
The Treasure's Code
Based on your final position, what are the Modulus and the Argument of the treasure's location? Show your work.
Length (Modulus)
Direction (Argument)
Mission Complete • Vector Voyages • 2026
