Seahorse Series · Quiz 3 · Addition and Subtraction with Regrouping

The Guide

Before you begin

Addition joins amounts; subtraction measures the gap between them. If you did Quiz 2, you can already say which number is bigger; this quiz answers by how much. And the whole machinery runs on one trade you already make at every cash register: ten of one thing for one of the next thing up. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

Line up the families first

Column arithmetic only works when places stand under their own kind — ones under ones, tens under tens. The rule of thumb: push both numbers to the right edge and let them line up from there. Most "wrong answers" in addition are really right answers to a misaligned problem.

Carrying: trading up

Add one column at a time, from the right. If a column's total reaches ten or more, it has grown too big for its place — so ten of that place trade up for one of the next place, exactly the way ten one-dollar bills trade for a single ten. The one ten you traded for is written small above the next column: that's the carry.

Fourteen ones make the trade: ten of them are bundled into a single ten, four stay behind. Write the 4, carry the 1.

Borrowing: trading down

Subtraction runs the same trade backward. Working a column, if the top digit is smaller than the one below it, you can't take away more than you have. So you break a ten: the next place over gives up one of its own, and it arrives in your column as ten. It's making change: the drawer has no ones, so you break a ten at the register.

One ten breaks into ten ones and joins the three already there: thirteen — now there's enough to subtract from.

Why "borrowing" is a bad name for a fair trade: nothing is ever paid back. The ten didn't lend — it became ten ones, the way a ten-dollar bill becomes ten singles at the register. The total never changed for a moment. Some books say regrouping instead, and that's the honest word for both directions: same amount, new grouping.

When the neighbor's pocket is empty

Sometimes you reach next door to break a ten and find a zero living there. Then the change-making runs down the street: the zero takes one from its own neighbor first, which makes it ten, and then it breaks one of those for you. In 5,003 − 1,247, the ones need help, the tens have nothing, the hundreds have nothing — so the trade starts all the way at the thousands and cascades down. You'll see it worked in full in Example 3. It's like a car's odometer rolling backward from 5,000 to 4,999: every zero turns into a 9.

Two habits that catch nearly every error

The guard is an estimate: before you trust an answer, round the numbers you started with and check that the answer lands nearby, just as in Quiz 2. The round trip: addition and subtraction undo each other — so check any subtraction by adding your answer back to what was taken away. If you don't land exactly where you started, something slipped.

Worked Examples

Three to study before you start

Example 1 · Addition, carrying three times

Add 3,478 + 2,846. Guard first: about 3,000 + 3,000, so the answer should live near 6,000.

	1	1	1
	3,	4	7	8
+	2,	8	4	6
	6,	3	2	4

Right to left: 8 + 6 = 14 — write 4, trade ten ones for one ten, carry the 1. Then 7 + 4 + 1 = 12 — write 2, carry 1. Then 4 + 8 + 1 = 13 — write 3, carry 1. Then 3 + 2 + 1 = 6. Answer: 6,324 — right in 6,000's neighborhood, as the guard predicted.

Example 2 · Subtraction, breaking tens

Subtract 852 − 367. The ones come up short, because you can't take 7 from 2, so break a ten. Then the tens come up short, so break a hundred.

	7	14	12
	8	5	2
−	3	6	7
	4	8	5

12 − 7 = 5, then 14 − 6 = 8, then 7 − 3 = 4: 485. The round trip proves it: 485 + 367 = 852, back where we started.

Example 3 · The cascade — change-making down the street

Subtract 5,003 − 1,247. The ones need help; the tens hold zero; the hundreds hold zero. So the thousands break first, and the trade cascades: 5 thousands become 4 thousands and ten hundreds; one hundred breaks into ten tens; one ten breaks into ten ones.

	4	9	9	13
	5,	0	0	3
−	1,	2	4	7
	3,	7	5	6

13 − 7 = 6, then 9 − 4 = 5, then 9 − 2 = 7, then 4 − 1 = 3: 3,756. Round trip: 3,756 + 1,247 = 5,003. The zeros didn't block the trade — they just passed it along, like the odometer rolling in reverse.

The Quiz · Ten Questions

Now you

Work without a calculator, in columns, on paper. Write your answers on the lines. Questions 3 and 8 are multiple choice — choose the one best answer.

Add: 4,213 + 2,564
Add: 356 + 528
Add: 4,687 + 2,935
- A) 6,512
- B) 7,622
- C) 6,622
- D) 7,522
Add: 847 + 92
Subtract: 968 − 435
Subtract: 7,241 − 3,586
Subtract: 6,005 − 2,478

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes with the leafy and weedy seadragons, filmed by Wired.

Denise worked 623 − 158 and got 535. What went wrong?
- A) She forgot to carry.
- B) In each column she subtracted the smaller digit from the larger, whichever was on top, instead of breaking a ten.
- C) She lined the numbers up wrong.
- D) Nothing: 535 is correct.
Rosa's take-home pay is $2,318 a month, and her rent is $1,475. How much of her pay remains after rent?
Marcus's odometer read 49,862 before a delivery run and 50,341 after. How many miles was the run?

Answer Key

Check your work

Open the key — after you've finished all ten

1 6,777

No column reaches ten, so no trades — pure facts and alignment. If this one went wrong, the trouble isn't regrouping yet; it's either a slip in a basic addition fact, or the columns weren't lined up.

2 884

6 + 8 = 14: write 4, carry 1. Then 5 + 2 + 1 = 8, and 3 + 5 = 8. One trade, cleanly made. If you got 874, the carried ten never arrived at the tens column.

3 B: 7,622.

Every wrong choice is a specific dropped trade. A) 6,512 is what the columns say if no carry is ever made. C) 6,622 makes every trade but the last — the final carry into the thousands went missing. D) 7,522 dropped just the middle one, the carry into the hundreds. If you picked any of them, you now know exactly which column to watch.

4 939

The setup is the whole question: 92 slides to the right edge, its 9 standing under the 4 (tens under tens), its 2 under the 7. If you got 1,767, the 92 drifted left and became 920 — right arithmetic, wrong alignment.

5 533

Every top digit is big enough; no tens get broken. Like question 1, this is the baseline — trouble here is facts or setup, not regrouping.

6 3,655

Three tens broken in a row: 11 − 6 = 5, then 13 − 8 = 5, then 11 − 5 = 6, then 6 − 3 = 3. Round trip: 3,655 + 3,586 = 7,241. If your digits differ in the middle, then after breaking a ten you forgot to lower the digit you broke it from: the struck-out digit kept its old value.

7 3,527

The cascade in full: 6,005 holds zeros in both the tens and the hundreds, so the ones' request for change travels all the way to the thousands. The 6 becomes 5. Ten hundreds arrive; one of them goes on to the tens, so nine hundreds remain. Ten tens arrive; one of them goes on to the ones, so nine tens remain. The ones become 15. Then 15 − 8 = 7, 9 − 7 = 2, 9 − 4 = 5, 5 − 2 = 3: 3,527. Round trip: 3,527 + 2,478 = 6,005. Exactly Example 3's street, one house over.

8 B.

The true answer is 465: break a ten (13 − 8 = 5), break a hundred (11 − 5 = 6), then 5 − 1 = 4, and the round trip confirms it: 465 + 158 = 623. Denise instead took the smaller digit from the larger in every column, whichever was on top: 8 − 3, 5 − 2, 6 − 1. It feels tidy, it never requires breaking anything, and it is always wrong the moment a top digit comes up short. Her tens digit gives it away: 3 where the true digit is 6. The rule beneath the rule: the top number is the one being taken from — when its digit is too small, it doesn't swap places; it breaks a ten.

9 $843

2,318 − 1,475: break a hundred, break a thousand: 843. Guard: about $2,300 − $1,500 = $800 — the neighborhood agrees. Round trip: 843 + 1,475 = 2,318. This is the subtraction most worth knowing by heart: what's left after the biggest bill is paid.

10 479 miles

50,341 − 49,862 = 479, a subtraction straight across the odometer's rollover. Guard — and here's the judgment call: rounding both to the nearest thousand gives 50,000 − 50,000 = 0, which guards nothing. Round to the nearest hundred instead: 50,300 − 49,900 = 400 — now the neighborhood speaks. When two numbers stand close together, estimate with a finer landmark. Round trip: 49,862 + 479 = 50,341.

Reading your results

Questions	The skill they test	If they gave trouble
1, 5	Facts and column setup	Slow, careful columns — the regrouping machinery isn't the issue yet.
2, 3	Carrying	Reread "Trading up" — every carry is ten-for-one at the register, and every carry must arrive next door.
4	Lining up the places	Push both numbers to the right edge; ones under ones, always.
6, 7	Borrowing, including across zeros	Break a ten; when the neighbor holds zero, let the change-making run down the street as in Example 3.
8	Spotting the flipped column	The smaller-from-larger habit — name it to tame it; the top number is the one being taken from.
9, 10	Regrouping in the world	Guard with an estimate, then make the round trip: answer plus what was taken should land you home.

Eight or more right: the trades are yours — move on to Quiz 4. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news — we've found the right counter to work at. Walk the two trade pictures slowly with your teacher, make change out loud, then come back to these same ten.

The Company · An Interlude

Fellow travelers

Our masthead seahorse — by all appearances Hippocampus kuda, and clearly a supporter of The People's Share — keeps the flag. But the family is large, and from time to time relations will visit the bottom of these quizzes. Three arrive today.

The pot-bellied seahorse (Hippocampus abdominalis) of southern Australia and New Zealand — largest of the whole tribe, tail curled like a treble clef.

A pot-bellied seahorse clinging to eelgrass, its brood pouch full and glowing pale in blue aquarium light — A father with a full pouch. Among seahorses, the males do the carrying — he holds the brood until the young are ready. Fitting company for a quiz about carrying.

A leafy seadragon drifting against a dark background, its body trailing ornate leaf-shaped appendages in violet and gold — The leafy seadragon (*Phycodurus eques*) — the cousin who went full ornament. This is the one from the rest stop's film; the weedy seadragon joins us another day.

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