Squaring Numbers Ending in 5

Mental Math · Axiom Academy

LESSON Squaring Numbers Ending in 5 Multiply the front part by the next number up, write 25 after it — and see exactly why that 25 is always there. 1. Take n, Then the Next Number Up Take 35² . Cover the 5 and look at what is left: 3 . That is n . Multiply it by the next number up — 3 × 4 = 12 — and then write 25 after it. The answer is 1225 , and you never touched 35 again. Every square of a number ending in 5 has this same shape: a computed front half, and a tail of 25 that was decided before you started. A computed head, then a tail that never changes 2. Why It Works: Grow the Square, Keep the Corner Draw 35² as a real square, 35 on a side, and cut it at 30 and 5. Four pieces fall out: a 30 × 30 block, two 30 × 5 strips , and a 5 × 5 corner . The whole trick lives in one observation — because this is a square , those two strips are identical . Stack them and they become a single strip 30 wide and 10 tall. Slide it under the block and the square has grown into a 30 × 40 rectangle : in tens that is 3 by 4 , which is n by n + 1 , so its area is twelve hundreds, or 1200. The corner never joined in. It sits out as 25 . And 1200 + 25 = 1225 . n tens by n tens. On its own it is 900, and it is the part of the answer that depends on n. Equal, because the two sides of a square are equal. Stacked, they form one strip 30 wide and 10 tall. Block plus stacked strip. One extra ten of height turns 3 tens into 4 tens: n becomes n + 1.

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