Vector Vibes Slides Vector Vibes
Geometric Addition & Subtraction
Unit: Fractal Frontiers | Lesson 01
The Flight Path Problem
Imagine you are a pilot. Your position is a point on a map.
Mission Parameters:
Start at \( 3 + 2i \)
Move by the vector \( 1 - 4i \)
Where do you land?
Complex numbers aren't just values; they are movements.
3 + 2i End: 4 - 2i
The Parallelogram Rule
Adding \( z_1 + z_2 \)
If you represent two complex numbers as arrows from the origin:
Place the tail of \( z_2 \) at the head of \( z_1 \).
The resulting vector points to the sum.
This forms a parallelogram!
(a + bi) + (c + di) = (a+c) + (b+d)i
z1 z2 z1 + z2
Algebraic addition matches geometric translation.
Subtracting Space
To subtract \( z_2 \), we add the opposite of \( z_2 \).
\( z_1 \)
Initial Position
\( z_2 \)
The Subtrahend
\( z_1 - z_2 = z_1 + (-z_2) \)
Geometrically, this means flipping the second arrow 180° and then adding it head-to-tail.
Quick Sketch
If \( z_1 = 2 + i \) and \( z_2 = -1 + 3i \):
1. Calculate \( z_1 + z_2 \) algebraically.
2. Visualize the path on a grid. Where do you end up?
3. Does the result match your calculation?
Grab your "Plane Paths" worksheet to try this now!
Plane Paths Worksheet Plane Paths
Complex Addition & Subtraction
Student Name
Date
Fractal Frontiers | Lesson 01
Part 1: The Parallelogram Rule
Graph the two complex numbers as vectors starting from the origin. Use the head-to-tail method to find their sum, then verify your answer algebraically.
1. Let \( z_1 = 3 + 2i \) and \( z_2 = 1 - 4i \)
Algebraic Work:
Final Sum:
Complex Plane
Re Im
Part 2: Vector Subtraction
To subtract, remember that \( z_1 - z_2 = z_1 + (-z_2) \). Flip the subtrahend 180° through the origin before adding.
2. Calculate \( (-2 + 3i) - (1 + 2i) \)
Algebraic Work:
Final Difference:
Complex Plane
Re Im
Part 3: Navigation Challenges
"A drone starts at position \( 5 - i \). It receives two displacement commands in the form of complex numbers: \( d_1 = -3 + 4i \) followed by \( d_2 = 2 - 5i \)."
A. What is the drone's final position? Show your step-by-step vector addition.
B. If the drone needs to return to the origin \( (0 + 0i) \) in a single flight, what complex number represents that return vector?
Conceptual Bridge
In your own words, why does adding the real parts and the imaginary parts separately work for vector translation? How does this connect to the x and y movements on a standard coordinate grid?
Vector Vision Teacher Guide Vector Vision
Teacher Facilitation Guide
Lesson 01 | Fractal Frontiers
Duration
50-60 Minutes
Focus Skill
Geometric Addition
Materials
Slides, Worksheet, Graph Paper
Instructional Sequence
1
The Flight Path Hook (10 min)
Open the slide deck to the Flight Path problem. Ask students to consider how a pilot might describe their movement. Does the pilot care about absolute coordinates, or their change in position?
Key Discussion Prompt:
"If I say 'Go North 3, East 2', how is that similar to the complex number \( 2 + 3i \)? Why does it matter where I start?"
2
Vectors & Parallelograms (15 min)
Explain that every complex number \( a + bi \) can be viewed as a vector from the origin to the point \( (a, b) \). Use the slides to demonstrate the "Head-to-Tail" method.
Addition: Slide the tail of vector 2 to the head of vector 1.Subtraction: Emphasize that \( A - B \) is just \( A + (-B) \). Reflect vector B across the origin before adding.
3
Worksheet: Plane Paths (20 min)
Hand out the "Plane Paths" worksheet. Circulate and check for common errors.
Watch For:
Students forgetting to move the second vector to the head of the first. They often draw both from the origin and get confused about where the sum is.
4
Closure: Conceptual Bridge (5 min)
Review Part 3 of the worksheet. Ask a student to share their "Return Vector" for the drone. Lead into the next lesson: "If addition is translation, what could multiplication be?"
Differentiation & Support
Scaffolding
Provide physical "vector strips" (arrows made of cardstock) that students can physically move on a large desk-sized coordinate grid.
Extension
Challenge students to find the magnitude (length) of the resultant vector using the Pythagorean theorem, foreshadowing the modulus of complex numbers.
Spin Shift Slides Spin Shift
The Geometry of Multiplication
Unit: Fractal Frontiers | Lesson 02
What happens when we multiply by \( i \)?
Let's track a point:
Start at \( z = 2 \)
\( 2 \times i = 2i \)
\( 2i \times i = -2 \)
\( -2 \times i = -2i \)
\( -2i \times i = 2 \)
Look at the pattern. It's not just numbers; it's a rotation .
2 2i -2 -2i
The Dual Action
Multiplying by any complex number
Rotation
It rotates the original vector by the angle (argument) of the multiplier.
Scaling
It scales the length (magnitude) of the original vector by the magnitude of the multiplier.
Spirals of Multiplication
What happens if we keep multiplying by a complex number \( z \) where \( |z| > 1 \)?
If you rotate and grow repeatedly, you get an outward spiral.
If you rotate and shrink (\( |z| < 1 \)), you get an inward spiral.
Iterative Growth Pattern
Ready to Rotate?
Open your "Rotation Reels" worksheet.
z_next = z_current × (multiplier)
Graph it
Calculate it
See the pattern
Rotation Reels Worksheet Rotation Reels
The Geometry of Multiplication
Student Name
Date
Fractal Frontiers | Lesson 02
Part 1: Spinning with \( i \)
Starting with \( z_0 = 4 + 0i \), multiply by \( i \) repeatedly and plot each subsequent point on the grid.
Iteration Calculation Result (\( z_n \)) \( z_0 \) Starting Point \( 4 + 0i \) \( z_1 \) \( (4 + 0i) \times i \) \( z_2 \) \( z_1 \times i \) \( z_3 \) \( z_2 \times i \) \( z_4 \) \( z_3 \times i \)
Observation:
Describe the visual movement of the points on the grid. What is the angle of rotation between each step?
Re Im
Part 2: The Spiral Challenge
Now, let's multiply by a number that isn't just \( i \). Let \( z_{n+1} = z_n \times (1 + i) \). Calculate the next three points starting from \( z_0 = 1 + 0i \).
\( z_1 = (1 + 0i) \times (1 + i) = \_\_\_\_\_\_\_\_ \)
\( z_2 = z_1 \times (1 + i) = \_\_\_\_\_\_\_\_ \)
\( z_3 = z_2 \times (1 + i) = \_\_\_\_\_\_\_\_ \)
Critical Thinking:
Compare the length (magnitude) of \( z_0 \) to \( z_3 \). Is the point moving away from or toward the origin? Why?
Trace the Spiral
Part 3: Predict the Path
If you multiply a complex number by \( 0.5i \) repeatedly, what will happen to the points on the graph over time?
Describe the movement...
Describe the final destination...
Spin Secrets Teacher Guide Spin Secrets
Teacher facilitation Guide
Lesson 02 | Fractal Frontiers
Focus
Multiplication
Geometry
Rotation/Scaling
Level
Conceptual Inquiry
Next Up
Iteration
The "Aha!" Moment
The core goal of this lesson is for students to see that multiplication in the complex plane is fundamentally different from multiplication on a 1D number line. On a line, it's just stretching; in the plane, it's a "spin and stretch."
1. The \( i \) Investigation (15 min)
Use the Slide 2 diagram. Don't tell them it's a 90° rotation yet. Ask them to calculate the sequence \( 2, 2i, -2, -2i, 2 \). Let them discover the square shape it makes on the grid.
Check for understanding: "If \( i \) is 90°, what would multiplying by \( -1 \) be? (180°)"
2. Spirals and Growth (20 min)
Transition to Slide 3. Introduce the idea that if the magnitude (length) of our multiplier is exactly 1 (like \( i \)), the point stays on a circle. If it's greater than 1, it spirals out.
Mathematical Connection: Help students link the algebraic FOIL method of \( (a+bi)(c+di) \) to the geometric outcome. Even though the algebra is messy, the geometry is elegant.
3. Facilitating "Rotation Reels" (15 min)
In Part 2 of the worksheet, students will calculate \( (1+i)^n \). This is a vital preparation for Lesson 4 (Mandelbrot).
\( z_1 = 1+i \)
\( z_2 = 2i \)
\( z_3 = -2 + 2i \)
Common Pitfall: Students often miscalculate \( i^2 \) as 1 instead of -1 during FOIL, which breaks the rotation logic.
Questions for Depth
"If we multiply by a number that is purely real (no imaginary part), what is the 'angle' of rotation? Why?"
"How can you look at a complex number \( a+bi \) and guess if it will make a sequence spiral inward or outward?"
Loop Logic Slides Loop Logic
Mastering Recursive Functions
Unit: Fractal Frontiers | Lesson 03
The Feedback Loop
Iteration is the process of repeating a function where the output of one step becomes the input of the next.
Input (z)
z
f(z)
Output
z'
Example: Square it!
Function: \( f(z) = z^2 \)
Start: \( z_0 = 1 + i \)
Iteration 1: \( (1+i)^2 = 1 + 2i - 1 = 2i \)
Iteration 2: \( (2i)^2 = -4 \)
Iteration 3: \( (-4)^2 = 16 \)
This sequence is heading to infinity!
The Secret Formula
Mandelbrot Iteration
\( z_{n+1} = z_n^2 + c \)
"The next Z equals the current Z squared, plus a constant C."
\( z \)
The evolving position
\( ^2 \)
Rotate and Scale
\( + c \)
The Offset (Shift)
How far is too far?
To understand fractals, we track the magnitude (distance from origin) of \( z \).
Distance Formula (Complex):
\( |a + bi| = \sqrt{a^2 + b^2} \)
Just like the Pythagorean Theorem!
The Boundary Rule:
If the magnitude ever exceeds 2 , the sequence will "escape" to infinity.
SAFE ZONE (Radius 2) ESCAPE!
Function Machine Time
We are going to iterate \( z^2 + c \) manually.
Your Goal
Calculate 3 steps of a sequence. If your value hits a magnitude of 2, you're an "escapee." If not, you're "trapped."
Toolbox
FOIL (Squaring Binomials)
Complex Addition
Distance Formula
Iteration Island Activity Iteration Island
Recursive Function Training
Student Name
Date
Fractal Frontiers | Lesson 03
The Iteration Engine: Squaring \( z \)
Before we iterate, let's practice the core step: squaring a complex number. Remember: \( (a + bi)^2 = a^2 + 2abi - b^2 \).
A. Square \( z = 1 + i \)
B. Square \( z = 2 - 2i \)
Part 2: The \( z^2 + c \) Loop
Starting with \( z_0 = 0 + 0i \), calculate the next three terms using the formula \( z_{n+1} = z_n^2 + c \). For each result, calculate its magnitude \( |z| \).
Challenge Case: \( c = 0.5 + 0.5i \)
| Step | Iteration Calculation | Magnitude \( |z| \) |
| --- | --- | --- |
| \( z_0 \) (Initial) | 0 + 0i | 0 |
| \( z_1 \) | \( (0 + 0i)^2 + (0.5 + 0.5i) = \)
| |
| \( z_2 \) | \( (z_1)^2 + (0.5 + 0.5i) = \)
| |
| \( z_3 \) | \( (z_2)^2 + (0.5 + 0.5i) = \)
| |
Divergence Check
Based on your calculation for \( z_3 \), is this sequence likely to diverge (go to infinity) or stay bounded ?
The Escape Rule
In the Mandelbrot set, if \( |z_n| > 2 \), the point \( c \) is NOT in the set.
Does \( |z_3| \) exceed 2? ______
Prediction: Safe or Gone?
Feedback Loop Teacher Guide Feedback Loop
Teacher facilitation Guide
Lesson 03 | Fractal Frontiers
Topic
Recursive Iteration
Formula
z^2 + c
Complexity
Algebraic High
Core Concepts
This lesson is the bridge between simple arithmetic and the infinite complexity of fractals. Students often struggle with the recursive nature of the work—losing track of which result is the "new input."
Key Teaching Points
1. The "Function Machine": Use the analogy of a factory line. One person squares the number, the next adds the constant \( c \), and then the result is sent back to the start of the line.
2. Squaring Binomials: Revisit FOIL. Remind students that \( (a+bi)^2 \) is NOT \( a^2 + (bi)^2 \). The middle term \( 2abi \) is where the rotation "logic" lives.
3. The Threshold: Why 2? Explain that for the formula \( z^2 + c \), once the distance from the origin is greater than 2, the "squaring" part of the formula becomes so large that adding \( c \) can no longer pull it back toward the center.
Common Misconceptions
Magnitude vs Coordinates: Students might think they "escaped" if a single coordinate is > 2. Remind them it's the distance from origin (\( \sqrt{a^2+b^2} \)).Replacing C: Some students accidentally change the constant \( c \) in each step. Emphasize that \( c \) is fixed for the entire sequence; only \( z \) changes.
Classroom Management: The Human Iteration
To make this concrete, have 3 students stand at the board.
Escape Velocity Slides Escape Velocity
The Divergence Test
Unit: Fractal Frontiers | Lesson 04
The Two Fates of Iteration
The Prisoner
Points that stay close to the origin forever. These points are Inside the Mandelbrot Set.
Magnitude \( |z_n| \) remains \( \le 2 \).
The Escapee
Points that fly away to infinity. These points are Outside the Mandelbrot Set.
Magnitude \( |z_n| \) eventually \( > 2 \).
Speed of Divergence
In fractal art, we don't just care if a point escapes, but how fast it escapes.
Fast Escape = Bright Color
Slow Escape = Darker Color
Never Escape = Black (The Set)
The Escape Count Table
Iter. Value Status 1 1.2 + 0.5i OK 2 2.1 - 0.2i ESCAPED!
This point has an "Escape Velocity" of 2 iterations.
The Visualization
Every pixel on your screen represents a complex number \( c \).
Computers run the \( z^2 + c \) loop thousands of times per pixel to see how fast they "fly away."
Your Task Today:
You are the computer. You will calculate the "Escape Value" for specific points assigned to your sector.
Prisoner or Escapee?
Get your "Prisoner or Escapee" lab sheet ready.
1
Pick a Coord
2
Iterate (3-5x)
3
Check for Escape
Prisoner Or Escapee Lab Prisoner or Escapee
The Mandelbrot Divergence Lab
Researcher
Date
Fractal Frontiers | Lesson 04
Lab Mission
"You are testing coordinate points to see if they belong to the Mandelbrot Set. If a point's magnitude exceeds 2, it 'escapes' and is not in the set. If it stays within the boundary after 5 iterations, it is considered a prisoner (for now)."
TEST POINT A: \( c = 0.3 + 0.2i \)
Iteration Target: 5
| Iteration | Result (\( z_n \)) | \( |z| \) |
| --- | --- | --- |
| z0 | 0 | 0 |
| z1 | | |
| z2 | | |
| z3 | | |
| z4 | | |
| z5 | | |
Final Determination:
PRISONER
ESCAPEE
If it escaped, at which iteration did \( |z| \) cross 2.0?
TEST POINT B: \( c = 0.5 + 0.6i \)
Iteration Target: 5
| Iteration | Result (\( z_n \)) | \( |z| \) |
| --- | --- | --- |
| z0 | 0 | 0 |
| z1 | | |
| z2 | | |
| z3 | | |
| z4 | | |
| z5 | | |
Final Determination:
PRISONER
ESCAPEE
If it escaped, at which iteration did \( |z| \) cross 2.0?
Post-Lab Analysis
Compare Point A and Point B. One of these points belongs to the Mandelbrot Set and the other does not. Why is it impossible to 100% prove a point is a "prisoner" forever using only 5 iterations?
Divergence Defense Teacher Guide Divergence Defense
Teacher facilitation Guide
Lesson 04 | Fractal Frontiers
Success Metric
Identifying the Escape Point
Mathematical Load
Complex Arithmetic
Student Task
Manual Lab Calculations
Instructional Goal
Students finally apply the recursive skill from Lesson 3 to a "real" data set. They act as human processors, testing points to determine the "border" of the Mandelbrot set.
1. Modeling the First Iteration (10 min)
Before releasing students to the lab sheet, model the first two iterations of a point at the board. Choose a point that escapes slowly (e.g., \( c = 0.4 + 0.4i \)) so they can see the magnitude creep toward 2.
Pro Tip: Have students use calculators only for final square root calculations; they should do the binomial squaring by hand to reinforce complex number rules.
2. Prisoner or Escapee? (25 min)
Distribute the Lab Sheet. Monitor students as they work. Many will stop calculating once magnitude hits 2.1 or 2.2—confirm that they are allowed to stop! That is the "Escape Velocity."
Point A: Will likely stay bounded or escape very slowly. It is a "Prisoner."
Point B: Will escape quickly (Iteration 2 or 3). It is an "Escapee."
3. Philosophical Conclusion (10 min)
Ask the class: "If a point hasn't escaped after 5 steps, can we be 100% sure it's a prisoner?"
Fractal Finale Slides Fractal Finale
The Thumbprint of Infinity
Unit: Fractal Frontiers | Lesson 05
The Power of Pixels
No single person can map the Mandelbrot set. It requires collaboration.
1 Each of you has calculated the destiny of specific complex points.
2 We will now assign colors based on your "Escape Velocity."
3 When we put them together, the image emerges.
Sector 4
Low-Res Fractal Generation
The Escape Key
NEVER
The Set
5 STEPS
Very Slow
4 STEPS
Slow
3 STEPS
Fast
1-2 STEPS
Instant
"The beauty of a fractal is the border —the region where tiny changes in the starting point \( c \) create massive changes in destiny."
Into the Deep
Observe a high-resolution zoom of the Mandelbrot Set.
Every intricate swirl, every repetitive pattern, is the result of the exact arithmetic we have been doing by hand.
Unveil the Set
Collect your sector coloring sheet. Use your divergence data to color the grid.
Color
Assemble
Behold
Pixel Power Project Pixel Power
Mandelbrot Sector Mapping
Collaborator
Sector #
Fractal Frontiers | Lesson 05
Assignment Instructions
1. Use your results from the Escape Velocity Lab (and the additional points provided below).Divergence Key .
Divergence Key
Never Escaped: BLACK
Escaped at Step 5: D. RED
Escaped at Step 4: RED
Escaped at Step 3: PINK
Escaped at Step 1-2: WHITE
Additional Sector Data (Sector 5 Sample)
Pixel Coord (c) Escape Result Assigned Color \( -0.7 + 0.1i \) Prisoner (5 Steps) BLACK \( -0.1 + 0.9i \) Escaped Iteration 3 PINK \( 0.5 + 0.5i \) Escaped Iteration 2 WHITE \( -0.8 + 0.15i \)
|
Your Sector Grid
Each square represents one complex point from your data above.
1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4
Cut along the thick black border to prepare for assembly.
Infinity Map Teacher Guide Infinity Map
Teacher facilitation Guide
Lesson 05 | Fractal Frontiers
Final Objective
"To reveal the Mandelbrot set through collaborative data visualization, proving that abstract complex arithmetic produces universal beauty."
Duration
1-2 Class Periods
1. Preparations
Assigning Sectors:
Create a master grid on a large wall (poster paper works well). Divide it into 25-30 "Sectors" (4x4 squares). Each student or group will calculate the escape values for the 16 points in their sector.
The "Set" Center:
Ensure you assign points within the main "bulb" of the set (around \( c = -1 + 0i \)) to some students and points on the chaotic "fringes" (around \( c = 0.3 + 0.5i \)) to others.
2. Collaborative Coloring (The "Render")
As students finish their "Pixel Power" sheet, have them use markers or colored pencils that match the Divergence Key. Emphasize that Black points are the core of the set—they never escape.
Reflective Discussion Prompts:
"Look at the transition between colors. Is it smooth or jagged? Why do you think that is?"
"If we increased our grid to 10,000 pixels instead of 400, how would the image change?"
"How did your individual calculations contribute to this global pattern?"
The Closing Concept
End the sequence by showing a high-resolution Mandelbrot zoom video (many are available on YouTube, such as "Mandelbrot Fractal Zoom"). Explain that what they are seeing is the exact same math they did, just performed billions of times by a processor.
Algebra standard met: Operations with Complex Numbers
Art connection: Generative visualization & Symmetry
