Level 3 · Basic Multiplication
General 2-Digit Squaring
Square any two-digit number using the difference-of-squares identity — the same algebra your students learned, used as a mental shortcut.
The Identity
The algebraic identity N² = (N − d)(N + d) + d² holds for any d. Choosing d to be the distance from N to the nearest multiple of 10 makes (N − d) and (N + d) both multiples of 10 — and multiplying two multiples of 10 is easy.
Worked Example — Near a Round Number
41²
- d = 1 (distance to nearest 10, which is 40)
- Bracket: (41−1) × (41+1) = 40 × 42
- 40 × 42 = 1680
- Add d²: 1680 + 1² = 1681
Worked Example — Midway
46²
- Nearest 10 is 50; d = −4 (or distance = 4)
- Bracket: 42 × 50 = 2100
- Add d²: 2100 + 16 = 2116
Why It's Easier
Multiplying two numbers that are equidistant from a round number (like 40 × 42 or 42 × 50) is much faster than computing 41 × 41 or 46 × 46 directly, because one of the numbers is always a multiple of 10 and the other is small.
💡 Teaching Tip
This is also an excellent way to make algebra feel useful. Write the identity on the board first, then show that mental math is just applied algebra. Students are often surprised that a "hard" math concept makes calculation easier.