Column addition & subtraction
Column (standard) written addition & subtraction, up to 3–4 digits; estimate the result first & check with the inverse.
Column addition and subtraction are the standard written methods for adding or subtracting large numbers by lining the digits up in neat columns — ones under ones, tens under tens — and working one column at a time.
What it means
Every whole number is built from place value: the digit’s position tells you what it is worth. In 342, the 2 is two ones, the 4 is four tens (40), and the 3 is three hundreds (300). Column methods use this by stacking numbers so that matching places sit in the same column, then handling each column separately, always starting from the right (the ones).
Addition. Add each column. If a column’s total reaches ten or more, it is too big to write as a single digit, so you carry: keep the ones-part of that total in the column and move the extra ten (or hundred) into the next column to the left. Ten ones become one ten; ten tens become one hundred.
Subtraction. Subtract each column, top minus bottom. If the top digit is smaller than the bottom one, you cannot take away enough, so you regroup (also called borrowing): take one unit from the next column left — turning it into ten of the current column — then subtract. One ten becomes ten ones; one hundred becomes ten tens.
Two habits make the method reliable. First, estimate the answer before you start by rounding (e.g. 342 + 218 ≈ 340 + 220 = 560) so a wildly wrong result stands out. Second, check with the inverse: addition and subtraction undo each other, so if 342 + 218 = 560, then 560 − 218 should give back 342.
Worked examples
Addition with a carry (the small 1 is the carried ten):
¹
3 4 7
+ 2 8 5
-------
6 3 2Ones: 7 + 5 = 12 → write 2, carry 1. Tens: 4 + 8 + 1 = 13 → write 3, carry 1. Hundreds: 3 + 2 + 1 = 6.
Subtraction with regrouping:
4 ¹5
5 ⁶3
- 2 7 8
-------
2 7 5Ones: 3 − 8 won’t go, so regroup a ten: 13 − 8 = 5. Tens: now 4 − 7 won’t go, regroup a hundred: 14 − 7 = 7. Hundreds: 4 − 2 = 2.
Check by the inverse: 278 + 275 = 553. It matches, so the subtraction is right.
The generative-art connection
Carrying and borrowing are hard to believe until you see the trade. In Number Pieces, a number is a heap of base-ten tiles — hundred-squares, ten-strips, unit-dots. Adding two numbers means gathering all the tiles; every time ten unit-dots pile up, you physically swap them for one ten-strip. That swap is the carry. Subtraction runs the film backwards: break one ten-strip into ten dots so you have enough to remove. The arithmetic becomes a visible act of grouping and regrouping, and the tidy columns of the written method are just a shorthand for the same moves.
Common misconceptions
- Forgetting to carry, writing
347 + 285 = 5122by jamming each column’s full total in. The carried ten must move left, not stay put. - Always doing bigger-minus-smaller within a column, so
53 − 78in the ones becomes8 − 3 = 5. You must subtract top minus bottom; when the top is smaller, regroup instead. - Misaligned digits. Writing
342 + 5needs the5under the ones, not the hundreds. Line places up, not left edges.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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Use column addition to work out `3 456 + 2 875`. Estimate first by rounding both numbers to the nearest hundred, then check your answer.
Answer
Estimate: `3 500 + 2 900 = 6 400`. Exact: ones `6+5=11` (write 1, carry 1), tens `5+7+1=13` (write 3, carry 1), hundreds `4+8+1=13` (write 3, carry 1), thousands `3+2+1=6`. So `3 456 + 2 875 = 6 331`, close to the estimate of 6 400. Correct.
Art hook Draw two stacked rows of base-ten dot-columns on a canvas; when any column exceeds 9 dots, animate 10 dots merging and one dot sliding left into the next column — the carry made visible.
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Use column subtraction to work out `6 002 − 1 547`. This one needs regrouping across zeros — take your time.
Answer
Ones: `2−7` won't go; the tens and hundreds are 0, so regroup from the thousands across: `6 002` becomes `5` thousands, `9` hundreds, `9` tens, `12` ones. Then `12−7=5`, `9−4=5`, `9−5=4`, `5−1=4`. Answer `4 455`. Check: `4 455 + 1 547 = 6 002`. Correct.
Art hook Show the top number as a bar of 6 large tiles; when subtracting across zeros, animate one tile cascading-breaking into ten smaller tiles one place down, repeating through the zeros.
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A shop sold `1 208` tickets on Friday and `975` on Saturday. How many tickets in total, and how many more were sold on Friday than Saturday? Show your column working for both.
Answer
Total: `1 208 + 975 = 2 183` (ones 8+5=13 write 3 carry 1; tens 0+7+1=8; hundreds 2+9=11 write 1 carry 1; thousands 1+1=2). Difference: `1 208 − 975 = 233` (ones 8−5=3; tens 0−7 won't go, regroup a hundred: 10−7=3; hundreds now 1−9 won't go, regroup a thousand: 11−9=2; thousands 0). Check: `233 + 975 = 1 208`. So 2 183 in total, and 233 more on Friday.
Art hook Two vertical dot-towers (Fri, Sat) side by side; a third tower rises as their sum and a small gap-marker highlights the difference between the first two.
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Lena worked out `4 000 − 1 236 = 3 236` in a hurry. Without doing the full subtraction, use estimation to explain why she must be wrong. Then find the correct answer.
Answer
Estimate: `4 000 − 1 200 ≈ 2 800`, nowhere near 3 236, so hers is too big — she likely subtracted the smaller digit from the larger in some columns instead of top-minus-bottom. Correct: `4 000 − 1 236 = 2 764`. Check: `2 764 + 1 236 = 4 000`. Correct.
Art hook A number line where a marker jumps from 4000 back by 1236; shade the 'reasonable zone' around 2 800 so wrong answers land visibly outside it.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Line these up in columns and add: `234 + 152`. No carrying is needed here.
Answer
Ones `4+2=6`, tens `3+5=8`, hundreds `2+1=3`. Answer `386`.
Art hook Draw the two numbers as three columns of dots each; the answer row simply combines the dots column by column, no merging yet — the calm before carries.
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Add these two, remembering to carry when a column reaches ten: `128 + 46`. Careful lining up the ones!
Answer
Line 46 under the ones/tens: ones `8+6=14` (write 4, carry 1), tens `2+4+1=7`, hundreds `1`. Answer `174`.
Art hook Highlight the ones column in one colour and the tens in another so the learner sees 46 slotting under the right places, not the left edge.
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Subtract in columns (no regrouping needed): `579 − 243`.
Answer
Ones `9−3=6`, tens `7−4=3`, hundreds `5−2=3`. Answer `336`.
Art hook Two dot-rows; the subtracted dots fade out, leaving the difference glowing — a gentle first subtraction with nothing to break apart.
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This subtraction needs one regroup: `362 − 148`.
Answer
Ones `2−8` won't go, regroup a ten: `12−8=4`; tens now `5−4=1`; hundreds `3−1=2`. Answer `214`. Check: `214+148=362`.
Art hook Animate a single ten-strip splitting into ten unit-dots the moment the ones column runs short.
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Add three-digit numbers with two carries: `486 + 357`. Write the small carried digits above the columns.
Answer
Ones `6+7=13` (write 3, carry 1), tens `8+5+1=14` (write 4, carry 1), hundreds `4+3+1=8`. Answer `843`.
Art hook Show carried 1s hopping up-and-left between columns like little arcs, one per carry.
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Spot the mistake: a pupil wrote `528 + 274 = 7912` by writing each column's total in full (ones `8+4=12`, tens `2+7=9`, hundreds `5+2=7`). What went wrong, and what is the correct answer?
Answer
They forgot to carry — a two-digit column total can't sit in one column. Correct: ones 12 (write 2, carry 1), tens `2+7+1=10` (write 0, carry 1), hundreds `5+2+1=8`. Answer `802`.
Art hook Side-by-side canvas: the buggy version shows overflowing dots spilling out of a column; the fixed version merges ten and slides the carry left.
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Estimate first: round each number to the nearest hundred to guess `612 + 389`. Then find the exact total and compare.
Answer
Estimate `600+400=1 000`. Exact: ones `2+9=11` (write 1, carry 1), tens `1+8+1=10` (write 0, carry 1), hundreds `6+3+1=10` → `1 001`. Very close to 1 000. Correct.
Art hook A dial/gauge points to the estimate 1000; when the exact answer resolves, a needle snaps to 1001 right beside it, showing the estimate was trustworthy.
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A four-digit addition with a carry: `2 617 + 1 495`. Estimate to the nearest thousand first.
Answer
Estimate `3 000+1 000=4 000`. Exact: ones `7+5=12` (2 carry 1), tens `1+9+1=11` (1 carry 1), hundreds `6+4+1=11` (1 carry 1), thousands `2+1+1=4`. Answer `4 112`.
Art hook Four dot-columns tall; carries ripple leftward in a chain, each triggering the next — a small cascade.
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Subtract with regrouping across two columns: `803 − 267`.
Answer
Ones `3−7` won't go; the tens are 0, so regroup from the hundreds: `803` → 7 hundreds, 9 tens, 13 ones. `13−7=6`, `9−6=3`, `7−2=5`. Answer `536`. Check: `536+267=803`.
Art hook Watch a hundred-tile break into ten ten-strips, then one of those break into ten dots — a two-step cascade across the zero.
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Reasoning: without fully calculating, is `3 128 + 2 954` closer to `5 000`, `6 000`, or `7 000`? Explain, then work it out.
Answer
Round: `3 000+3 000=6 000`, so it's closest to 6 000. Exact: `3 128+2 954=6 082`. Yes, nearest to 6 000.
Art hook Three target rings labelled 5000/6000/7000; a dot flies toward the ring your estimate picks, then lands exactly at 6082.
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Use the inverse to check: someone claims `5 214 − 2 876 = 2 338`. Add to check whether the answer is right.
Answer
Check: `2 338 + 2 876 = 5 214` (ones 8+6=14, tens 3+7+1=11, hundreds 3+8+1=12, thousands 2+2+1=5). It gives back 5 214, so the subtraction is correct.
Art hook An arrow loops right for the subtraction and back-left for the addition, forming a circle that 'closes' when the check returns to the start number.
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Missing digit puzzle: in the column sum `4 _ 6 + 3 8 5 = 8 5 1`, what digit goes in the blank tens place of the first number?
Answer
Ones `6+5=11` (write 1, carry 1). Tens: `? + 8 + 1` must end in 5 with a carry, so `?+9=15`, giving `?=6`. Check hundreds `4+3+1=8`. The blank is `6`; the sum is `4 6 6 + 3 8 5 = 8 5 1`.
Art hook A grid with one empty cell that glows; dragging a dot-count into it updates the running column totals live until the sum turns green.