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The Seahorse Series · Math Diagnostics · Quiz 4

Multi-Digit Multiplication

Equal groups, the area of a rectangle, and addition that learned to move in bulk
The Guide

Before you begin

Multiplication is addition in bulk: 7 × 346 means seven 346s joined at once, without the tedium of writing 346 seven times. If you did Quiz 3, the carries and the ten-for-one trade at the register are back here too; if not, the Guide shows them as they come. And there is a picture that makes the whole operation honest: a rectangle. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

The toolkit

Column multiplication leans on the one-digit facts. If some are still shaky, that is nothing to fear: the toolkit is right here, and it's small enough to keep in your pocket:

Open the toolkit — the times table through 10
×12345678910
112345678910
22468101214161820
336912151821242730
4481216202428323640
55101520253035404550
66121824303642485460
77142128354249566370
88162432404856647280
99182736455463728190
10102030405060708090100

The table is wider than your screen — slide it sideways with your finger, or turn your phone.

The gold diagonal, running 1, 4, 9, 16, 25 and onward, is the perfect squares, each number times itself. Remember them; they return in a later quiz wearing fancier clothes, as exponents and square roots.

One digit at a time, carries riding along

To multiply 7 × 346, the 7 visits every place of 346 in turn, from the right, and the carries ride along exactly as in addition: 7 × 6 = 42 — write 2, carry 4. 7 × 4 = 28, plus the 4 riding along, 32 — write 2, carry 3. 7 × 3 = 21, plus 3, 24. Answer: 2,422.

Underneath, this is expanded form, from Quiz 1, doing the work: 7 × 346 is really 7 × 300 + 7 × 40 + 7 × 6 — that is, 2,100 + 280 + 42 = 2,422. The column method does the same thing in shorthand.

The rectangle that tells the truth

Here is multiplication's honest picture. To multiply 23 × 14, build a rectangle 23 across and 14 down, then cut it along place-value lines: 23 into 20 and 3, 14 into 10 and 4. Four rooms appear, and each room's area is one small multiplication:

So 23 × 14 = 322, and nothing was hidden: every part of each number met every part of the other, exactly once. Turn the rectangle on its side and nothing changes — which is why 14 × 23 is the same answer, always.

The standard method is the rectangle, folded up

Column multiplication computes the same rooms in two rows. First row: 23 × 4 = 92 — that's the bottom rooms, 80 + 12. Second row: 23 × 1 — but that 1 is a ten, so the row means 23 × 10 = 230, the top rooms, 200 + 30. The second row begins one place to the left because it is made of tens; some people write a 0 there to hold the seat. Then add: 92 + 230 = 322.

Why does the second row slide left? Because the digit driving it isn't what it looks like. In 14, the 1 is a ten — Quiz 1's oldest lesson — so its whole row is ten times bigger and moves up one place to say so. Forgetting the slide is treating a ten like a one, and the answer comes out roughly ten times too small. A quick estimate catches it instantly.

Tens and hundreds travel light

Multiplying by 10, 100, or 1,000 slides every digit up one, two, or three places — the zeros simply hold the seats left empty: 47 × 10 = 470, 47 × 100 = 4,700. This is why the second row trick works, and it makes estimates fast: 79 × 52 is about 80 × 50 = 4,000 — the estimate standing guard, as always, before you trust anything.

Worked Examples

Three to study before you start

Example 1 · One digit, carries riding

7 × 346. Guard: 7 × 350 is about 2,450.

34
346
×7
2,422

7 × 6 = 42, write 2 carry 4. 7 × 4 = 28 + 4 = 32, write 2 carry 3. 7 × 3 = 21 + 3 = 24. Answer 2,422 — the guard nods.

Example 2 · Two rows, one rectangle

23 × 14 by the standard method — watch each row claim its rooms from the figure above:

23
×14
92
230
+322

Row one: 23 × 4 = 92 — the bottom rooms (80 + 12). Row two: 23 × 10 = 230 — the top rooms (200 + 30), the parked 0 holding the ones seat because this row is made of tens. Sum: 322.

Example 3 · The guard at work

58 × 41. Guard first: about 60 × 40 = 2,400. Now compute: 58 × 1 = 58; 58 × 40 = 2,320; together 2,378. The answer lives right where the guard said it would. Had a slide gone missing, giving 58 + 232 = 290, the guard would have caught a number ten times too small before it left the page.

The Quiz · Ten Questions

Now you

Work without a calculator, in columns, on paper. Write your answers on the lines. Questions 4 and 7 are multiple choice — choose the one best answer.

  1. Multiply: 3 × 213
  2. Multiply: 6 × 478
  3. a) 47 × 10    b) 47 × 100
  4. Multiply: 36 × 24
    • A) 216
    • B) 864
    • C) 7,344
    • D) 844
  5. Multiply: 57 × 63
  6. Multiply: 208 × 34
Film frame: a leafy seadragon in luminous blue close-up, striped body and leaf-shaped fins

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

  1. Brian worked 45 × 32 like this: first row 90, second row 135, added them, and got 225. What went wrong?
    • A) The second row never slid left — the 3 in 32 is three tens, so that row is really 1,350, not 135.
    • B) A times-table slip: 45 × 2 isn't 90.
    • C) The numbers were lined up wrong from the start.
    • D) Nothing: 225 is correct.
  2. Round each factor and estimate: 79 × 52
  3. Diego earns $28 an hour and worked 36 hours this week. How much did he earn?
  4. A community garden bed measures 24 feet by 15 feet. How many square feet of soil does it hold?
Answer Key

Check your work

Open the key — after you've finished all ten
1 639
3 × 3 = 9, 3 × 1 = 3, 3 × 2 = 6 — no carries anywhere. The baseline: if this one wobbled, the work ahead is toolkit facts, not the method.
2 2,868
6 × 8 = 48, write 8 carry 4. 6 × 7 = 42 + 4 = 46, write 6 carry 4. 6 × 4 = 24 + 4 = 28. If your middle digits differ, a carry was dropped on the way.
3 a) 470   b) 4,700
Every digit slides up one place, then two; the zeros hold the vacated seats. If you wrote 4,070 for (b), the digits slid separately — they travel together, as a block.
4 B: 864.
36 × 4 = 144; 36 × 20 = 720; sum 864. The wrong answers each name their error: A) 216 comes from adding 144 + 72 — the second row never slid, a ten treated as a one. C) 7,344 slid the second row twice, making 7,200 of what should be 720. D) 844 dropped the carry inside 36 × 4 (writing 124 for 144). Your guard knew all along: 36 × 24 is about 40 × 20 = 800, so only B was ever in the neighborhood.
5 3,591
57 × 3 = 171; 57 × 60 = 3,420; together 3,591. Guard: 60 × 60 = 3,600 — a near miss on purpose, since 57 and 63 straddle 60 from both sides.
6 7,072
208 × 4 = 832; 208 × 30 = 6,240; sum 7,072. The zero inside 208 must hold its place in every row, as zeros always do: 4 × 0 = 0 and 3 × 0 = 0 hold their seats. If you got 952, the zero's place collapsed and 208 quietly became 28: 28 × 34 = 952. The guard would have caught it, since 200 × 30 alone is 6,000.
7 A.
Brian's facts were perfect, since 45 × 2 = 90 and 45 × 3 = 135, but the 3 in 32 is three tens, so its row means 1,350. Slid properly: 90 + 1,350 = 1,440. The warning was available for free: 45 × 32 must land near 45 × 30 = 1,350, and 225 isn't even close. The estimate guard is precisely the alarm for missing slides, because a lost slide shrinks an answer about tenfold.
8 About 4,000
79 rounds to 80, 52 rounds to 50, and 80 × 50 = 4,000. (The true product is 4,108 — the estimate landed close, and fast.) Anything from a sound rounding is right here; the skill being tested is reaching for the landmark numbers at all.
9 $1,008
28 × 36: 28 × 6 = 168; 28 × 30 = 840; together 1,008. Guard: 30 × 36 ≈ 1,080 — neighborhood confirmed. Note the zero inside the answer holding the tens seat: one thousand eight, not one hundred eight.
10 360 square feet
24 × 15 = 360 — and this one is the rectangle from The Guide, wearing soil. Area is why the picture tells the truth: a floor 24 across and 15 deep simply holds 24 × 15 unit squares. Cut it into 20 + 4 and 10 + 5 and check the four rooms if you like: 200 + 100 + 40 + 20 = 360.

Reading your results

Questions The skill they test If they gave trouble
1, 2 One digit at a time, carries riding The toolkit table, then slow columns — every carry must arrive next door.
3 Tens and hundreds Digits slide as a block; zeros hold the vacated seats.
4, 5, 7 The two-row method and the slide Reread "the rectangle, folded up" — the second row is made of tens and moves left to say so.
6 Zeros inside a factor Quiz 1's silent zeros, on duty in every row.
8 Estimation Round both factors to their landmarks; let the guard speak first.
9, 10 Multiplication in the world Wages are equal groups of hours; area is the rectangle itself.

Eight or more right: the machine is yours — on to Quiz 5. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news — we've found the right room to work in. Draw the rectangles by hand with your teacher, label every room, then come back to these same ten.

Where this goes

Next in the Seahorse Series: Quiz 5 — Division, and What a Remainder Means. Division is multiplication run backward — the fair-share question — and the leftover it sometimes hands you is not an error but an answer with a story.

The Company · An Interlude

Fellow travelers

Two supporters have joined the masthead above — the leafy seadragon to the left and, to the right, an opal seahorse that appears to be an artist's invention rather than any species the ocean has issued. That is entirely proper: the supporters flanking a coat of arms have always been imagined beasts. Down here, the promised guest arrives, with a cousin.

A weedy seadragon swimming among kelp, slender gold body with violet-striped neck and small translucent paddle fins
The weedy seadragon (Phyllopteryx taeniolatus) — promised in Quiz 3, delivered here: the leafy's leaner cousin, rigged with paddles instead of pennants. Both swim in the rest stop's film.
A mottled seahorse with a deep pale belly holding station against dark reef growth
A mottled seahorse holding station in the dark — patient, upright, unhurried. The posture this series recommends.
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