Scroll 2 · read, poke, then train
Grown-ups call them “properties of operations” — long words for simple tricks. Each one is a move that lets you rearrange numbers without changing the answer, so hard sums become easy ones. Ninjas don't work harder. They move smarter.
Highlight any sentence to ask Sensei ✨
Swap the two numbers around — the answer doesn't change. 3 + 5 and 5 + 3 both make 8. 4 × 6 and 6 × 4 both make 24. That's the commutative property (from “commute” — to move around).
⚠️ But careful: the Switcheroo works for adding and multiplying only. Try it on subtraction and it breaks — 12 − 5 is 7, but 5 − 12 is −7. Not the same!
When you add (or multiply) three numbers, you get to choose which pair to do first. The parentheses ( ) just mean “do me first!” — and moving them doesn't change the answer. That's the associative property (numbers picking who they “associate” with first).
Why care? Because some pairs are friendlier than others. In 7 + 9 + 1, doing 9 + 1 = 10 first is way easier than 7 + 9. Regroup, then relax.
This is the flashiest move. When a number multiplies a sum in parentheses, it can clone itself and multiply each part separately:
3 × (4 + 2) = 3 × 4 + 3 × 2
Both sides make 18 — count the dots below and see it with your own eyes. The distributive property is the bridge between multiplying and adding, and it's the move algebra uses constantly.
Two numbers have magic powers:
Zero is invisible to adding: n + 0 = n. Nothing happens.
That's the identity property of addition.
One is invisible to multiplying: n × 1 = n.
That's the identity property of multiplication.
And one bonus power: anything × 0 = 0. Zero doesn't just hide — in multiplication it swallows everything. 🕳️
Type any number and feed it to a machine.
| Ninja move | Real name | What it says | Works for |
|---|---|---|---|
| 🌀 Switcheroo | Commutative | a + b = b + a · a × b = b × a | + and × only |
| 🤝 Regroup | Associative | (a+b)+c = a+(b+c) | + and × only |
| 💥 Multiplier Clone | Distributive | a × (b + c) = a×b + a×c | × over + |
| 🪞 Do-Nothing | Identity | n + 0 = n · n × 1 = n | always |
Five questions about the secret moves. Answer in your own words — Sensei reads every answer, cheers when you nail it, and helps when you're stuck.