Zoom out: what does this whole Mental Math module actually rely on?

A handful of algebra facts — place value, the distributive law, binomial squares, and modular arithmetic — dressed up as fast, memorable shortcuts.

Everything here is standard arithmetic rearranged for the mind:

• Distributive law → all the multiplication and percentage tricks (split into easy pieces).
• Binomial square → every squaring shortcut.
• 10 ≡ 1 (mod 9) and 10 ≡ −1 (mod 11) → the divisibility rules and the casting-out checks.
• Constant difference / compensation → the addition and subtraction shortcuts.

No magic, no ancient secrets — just number sense you build by practising the pieces until they're automatic.