Teacher Guide

📥 Printables

About this course

This is a discovery course in algebraic structure — identity, commutativity, associativity, inverses, and distribution — taught with colored circles and Cayley tables before any numbers appear. Students who "hate x's" don't recognize this as algebra until Station 4, when the colors turn into numbers and the reveal lands: you've been doing algebra all along.

Format: 7 stations, each pairing a printable worksheet with a short video, plus a free-play Interactive Cayley Explorer. Roughly 15–25 minutes per station; run one per day, or two per week.

⚠️ The one rule that matters

Worksheet BEFORE video, every time. The worksheets are designed for productive struggle — students discover each property themselves in the color tables. The video then names and confirms the discovery. Reversing the order turns discovery into mere confirmation and quietly deletes most of the learning. Hold the line even when students ask to "just watch first."

🎭 Why this course has no subtraction or division

Keeping − and ÷ around teaches students that the properties are trivia for a test — because half of "their" operations don't obey them. This course treats subtraction and division as nicknames: x − 5 is x + (−5), and x ÷ 2 is x × ½. Step Zero of every problem is translating the nicknames away. Three things happen:

• The properties become load-bearing. Everything on the page really is commutative, associative, and distributive — so students see the properties driving the solving, not sitting in a box labeled "of addition and multiplication" that they never open.
• Sign errors become impossible. In x − 5(2 − x), students distribute a bare 5 and orphan the minus. Translated — x + (−5)(2 + (−x)) — the sign is glued to its number and travels with it. This is the #1 error source in all of algebra, gone.
• The PEMDAS myths retire. No more "multiply before divide" or "add before subtract" — those operations no longer exist. One rule survives (× before +), and commutativity handles the rest.

Insist on the language: students add the inverse (a 0 appears — the identity of addition) and multiply by the inverse (a 1 appears — the identity of multiplication). Same move, two operations. That parallel is the whole game.

Facilitation notes by station

• 0 · Power of Structure: no math yet — organization as a superpower. Sets the frame.
• 1 · One Operation: reading a Cayley table; identity ("the do-nothing color"), commutativity (mirror symmetry across the diagonal!), associativity, inverses.
• 2 · Two Operations: ⊕ and ★ together; watch for students noticing distribution before it's named.
• 3 · Inverses & Undo: the heart of equation-solving — every step has an undo.
• 4 · Colors to Numbers: the reveal. Let students have the "wait, this is just adding?!" moment out loud.
• 5 · New Mental Model: what to do when arithmetic intuition runs out — lean on structure.
• 6 · Let's Do Algebra: real equations, solved with named moves. Connect each step back to its station.

🧭 Companion units