Unit 2 · Topic 03 · Operations
Anaya multiplied 34 by 25 and got 238.
Her estimate said the answer should be around 900.
She had done both multiplications perfectly. She had just stacked them in the same column.
The problem was 34 × 25 — the cost of 25 notebooks at 34 rupees each.
Anaya set it out in columns. First she multiplied 34 by the 5: thirty-four fives are 170. She wrote 170.
Then she multiplied 34 by the 2: sixty-eight. She wrote 68 directly underneath, lined up on the right, the way she lined up an addition.
170 plus 68 is 238.
She checked it against her estimate — 30 times 25 is about 750, so the answer should be somewhere near 800 or 900. 238 was nowhere near.
"Read me the 2 in twenty-five," said Ms. Rao.
"Two."
"What is it worth?"
Anaya stopped. "Twenty."
"So when you multiplied by it, what did you actually work out?"
"Thirty-four twos." She saw it. "But I needed thirty-four TWENTIES."
"Which is ten times bigger. Sixty-eight is right — but it is sixty-eight TENS, so it has to be written as 680."
Anaya rewrote it. 170 on the first line. 680 on the second, shifted one place left with a 0 holding the ones column.
170 + 680 = 850.
"Put the zero in deliberately," said Ms. Rao. "Not as decoration. It is the reminder that this row is tens, not ones. When you go up to three digits, the third row gets two zeros, because it is hundreds."
Anaya tried 234 × 123 that evening. Row one: 234 × 3 = 702. Row two: 234 × 20 = 4680. Row three: 234 × 100 = 23400. Total 28,782.
Her estimate had said 200 × 100 = 20,000, so the answer being a bit over that felt right. She had started checking before she started calculating.
Long multiplication breaks one hard multiplication into several easy ones and adds them up. 34 × 25 becomes 34 × 5 plus 34 × 20.
Each row is multiplied by a digit, but that digit carries its own place value. The tens digit is not 2, it is 20 — so its row is ten times bigger and must be SHIFTED one place left. Writing a 0 in the ones column of that row makes the shift visible instead of remembered.
The third row, multiplied by the hundreds digit, shifts two places and takes two zeros. The pattern continues for as many digits as the multiplier has.
Estimating first gives you a safety net. Round both numbers, multiply the round ones, and you know roughly where the answer must land. A missing shift makes the answer many times too small, and the estimate catches it immediately.
Work out how many hours there are in a year by multiplying 24 by 365, using the column method.
Estimate it first as 25 × 400. Compare the two — your estimate should be a little larger, and you should be able to say why.
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