Algebra’s Hidden Structure
Discover why algebra works using colored circles and Cayley tables
Interactive explorations • Video lessons • Printable worksheets
What’s This About?
Most students learn algebra rules without understanding WHY they work. These activities reveal the hidden patterns that make algebra logical — using colors instead of numbers!
Interactive Cayley Table Explorer
Click colors to see how they combine! Watch animated demonstrations of each algebraic property. Perfect for teaching or self-exploration.
Best for: Teachers, students, self-learners | Time: Explore freely
Getting Started
Begin here to discover the power of structure and organization
Do the Worksheet 0 BEFORE watching the video: The Power of Structure
Learn to recognize the power of structure and organization before diving into the math.
Difficulty: Beginner | Time: 15-20 min
Video Intro
Watch this video after completing Worksheet 0 to see the big picture of what you’ll be learning.
Time: ~8 min
One Operation
Discover identity, commutativity, associativity, and inverses using a single operation
Video 1: One Operation
Discover the identity element, commutativity, associativity, and inverses using a single ⊕ operation.
Time: ~8 min
Worksheet 1: One Operation
Learn to read a Cayley table and discover four fundamental properties: identity, commutativity, associativity, and inverses.
Difficulty: Beginner | Time: 15-20 min
Two Operations
See how ⊕ and ★ work together — different identities, the zero property, and the distributive property
Video 2: Two Operations
See how ⊕ and ★ work together — different identities, the zero property, and the distributive property!
Time: ~10 min
Worksheet 2: Two Operations
Explore how ⊕ and ★ work together. Discover different identities, the zero property, and the distributive property.
Difficulty: Intermediate | Time: 15-20 min
Inverses and Undo
Learn what an inverse is, why it matters, and how to use inverses to undo operations
Video 3: Inverses
See what an inverse is and why it matters. Learn how to use inverses to undo things!
Time: ~10 min
Worksheet 3: Undo
Learn about inverses — the key to solving equations and undoing operations.
Difficulty: Intermediate | Time: 15-20 min
Colors to Numbers
Reveal the secret that subtraction and division break the rules algebra needs — and how to fix it
Video 4: Colors to Numbers
In this video, we reveal the secret that subtraction and division break the rules algebra needs — and how to fix this problem.
Time: ~10 min
Worksheet 4: Colors to Numbers
See how the color patterns connect to actual algebra with numbers.
Difficulty: Intermediate | Time: 20-30 min
New Mental Models
You’re not getting worse at math — your mental models are being upgraded
Video 5: New Mental Model
Learn why the old mental models that worked in earlier grades no longer hold for what you are learning now!
Time: ~10 min
Worksheet 5: When Your Old Math Tricks Quit Working
You’re not getting worse at math. Your mental models are being upgraded.
Difficulty: Intermediate | Time: 20-30 min
Let’s Do Algebra!
Now you’re ready to solve real algebraic equations
Video 6: Let’s Do Algebra!
In this video, we show how to solve algebraic equations using everything you’ve learned.
Time: ~10 min
Worksheet 6: Doing Real Algebra
Now you are ready for some real algebra. Put everything together!
Difficulty: Intermediate | Time: 20-30 min
What You’ll Discover
Identity Elements
The “do nothing” elements — like 0 for addition and 1 for multiplication
Commutativity
Why order doesn’t matter — and how to SEE it in the table’s symmetry
Associativity
Why grouping doesn’t matter — and why we can drop parentheses
Inverses
The “undo button” for each element — the key to solving equations
Zero Property
Why anything times zero equals zero — watch Black “swallow” everything!
Distributive Property
How multiplication and addition connect — the bridge between operations
For Teachers
These materials work for bright 5th-6th graders ready for abstract thinking, OR for older students who learned algebra procedures without understanding the underlying structure. The color-based approach strips away numerical preconceptions and makes abstract properties visible.
Suggested sequence: Work through each video/worksheet pair in order, then explore freely with the Interactive Explorer