Math · Stage 6

Near-base products using the difference of squares

Two numbers sitting the same distance each side of a round base? Multiply them in your head: square the base, square the gap, and subtract. One of mental maths' most elegant shortcuts — and your first taste of algebra.

✦ Math📘 Stage 6 · S6⏱ ~12 min🧒 Ages 10–12★ Free to start

I can multiply near-base numbers using the difference of squares.

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Kavi's Lightning Multiply ⚡

A friend dares Kavi: "Quick — what's 47 × 53?" Before anyone reaches for a pencil Kavi hoots "2491!" The secret? Both numbers sit 3 away from the round base 50. "Square the base, square the gap, subtract: 50² − 3² = 2500 − 9," Kavi grins. The two numbers are balanced around a base, so the messy middle melts away.

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What your child will learn

1Two numbers, balanced around a round baseThis shortcut works when the two numbers are the same distance each side of a round base (10, 20, 50, 100…). Find the base in the middle, then the gap — how far each number is from it. One number is base − gap, the other is base + gap. Tap a row.

2Square the base, square the gap, subtractHere is the magic. When the factors are base − gap and base + gap, their product is just base² − gap². Square the base (easy — it's round), square the little gap, and take one from the other. Tap a row to see it worked out.

3Why it works — your algebra seedSpread out (base − gap)(base + gap) and you get base² + base·gap − base·gap − gap². The two middle pieces are equal and opposite, so they cancel — leaving just base² − gap². That balance is exactly why the numbers must be the same distance from the base. Tap a row.

4The same move works for hundreds and thousandsPick the nearest round base, whatever its size. The bigger the base, the more impressive the shortcut — you turn a scary multiplication into one easy square minus a tiny one. Tap a row.

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A peek inside

Multiply 47 × 53 in your head.

Both are 3 from the base 50. So 50² − 3² = 2500 − 9.

Try this at home: Base 10: 8 × 12 = __ · 9 × 11 = __ · 7 × 13 = __

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

Start the free lesson →⬇ Download PDF

Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.