Multiplication — tables (1–20), long multiplication, area model

WHAT is multiplication?

$a \times b = b \times a \qquad \text{(commutative)}$

Tables 1–20 — the 80/20 that saves you

Formula Derivation tricks (build a fact from a nearby fact)

×10: append a zero. $7\times10=70$ (each group of 7, ten times = 7 tens). WHY: place value shifts one column left.
×9: $9=10-1$ , so $n\times9 = n\times10 - n$ . E.g. $7\times9 = 70-7=63$ .
×11 (n≤9): $n\times11 = n\times10+n$ , i.e. write the digit twice: $7\times11=77$ .
×5: half of ×10. $8\times5 = 80/2 = 40$ .
Doubling: ×2 then ×2 gives ×4; ×2 thrice gives ×8. $6\times8 = 6\times2\times2\times2 = 48$ .
Near a known square: $13\times12 = 13\times12$ ; use $12\times12=144$ , add one more $12$ : $144+12=156$ ? No — add one more 13: careful! $13\times12 = 12\times12 + 12 = 144+12=156$ . (Adding one more group of 12.)

Long multiplication — WHY it works

Worked example

$47 \times 23$ (two-digit multiplier) Split $23 = 20+3$ . Compute two partial products, then add.

     4 7
   × 2 3
   ------
   1 4 1   ← 47 × 3   (7×3=21 write1 carry2; 4×3=12+2=14)
 9 4 0     ← 47 × 20  (write a 0 first, then 47×2=94)
 ------
 1 0 8 1

Why the shift/zero on the second row? The 2 in 23 is 2 tens, so its partial product is $47\times20$ , which ends in a 0. The shift = multiplying by 10. Final: $47\times20 + 47\times3 = 940+141 = \mathbf{1081}$ .

Figure — Multiplication — tables (1–20), long multiplication, area model

Area model — the picture behind it all

Worked example

$47 \times 23$ by area model Split sides: $47 = 40+7$ , $23 = 20+3$ . Four cells:

×	40	7
20	$800$	$140$
3	$120$	$21$

Sum $= 800+140+120+21 = \mathbf{1081}$ . Why this matches long multiplication: $800+140 = 940 = 47\times20$ (top row) and $120+21 = 141 = 47\times3$ (bottom row). Same two partial products! The area model just splits them further.

Common mistakes (Steel-manned)

Active recall

Recall Explain to a 12-year-old (Feynman)

Multiplying big numbers is like tiling a floor. Chop the floor into a few neat rectangles (tens and ones), find the tiles in each small rectangle (easy!), then add them up. You never multiply the whole scary number at once — you multiply friendly chunks and add. "Long multiplication" is just this chunk-and-add written in a tidy column, and the little zero you add on the second line is there because that digit was really tens, not ones.

Recall Forecast-then-verify

Before computing $34\times12$ : forecast it's near $34\times12 \approx 30\times12=360$ plus a bit → guess ~400. Now verify: $34\times12 = 340+68 = 408$ . Close to forecast ✓ — good sanity check that you didn't misplace a zero.

What does

a \times b

mean as repeated addition?

a

copies of

b

added together (=

b

copies of

a

Why is

a\times b = b\times a

A dot grid of

a

rows ×

b

columns has the same dots counted by rows or columns (commutativity).

State the distributive law.

a\times(b+c) = a\times b + a\times c

Trick for

n\times 9

n\times10 - n

(e.g.

7\times9 = 70-7 = 63

Trick for

n\times 5

Half of

n\times10

(e.g.

8\times5 = 80/2 = 40

47\times23

, why does the second partial-product row start with a 0?

The 2 is 2 tens, so that row is

47\times20

, which ends in 0.

What is a partial product in the area model?

The area of one cell = one place-value chunk of one factor times a chunk of the other.

Compute

47\times23

and give the four cell areas.

800+140+120+21 = 1081

Fix for the error

a(b+c)=ab+c

a

multiplies every term:

ab+ac

16\times15

by splitting?

16\times10 + 16\times5 = 160+80 = 240

Connections

Addition — carrying and place value (multiplication reduces to repeated addition)
Place Value & Number Systems (why the placeholder zero works)
Distributive Law (engine of every multiplication algorithm)
Division — inverse of multiplication
Squares & Square Roots (tables 1–20 speed these up)
Algebra — Expanding Brackets (area model → $(x+a)(x+b)$ )

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Multiplication ka matlab hai repeated addition — $4\times3$ yaani teen ko chaar baar jodo. Par har baar jodna slow hai, isliye hum tables (1–20) yaad karte hain taaki turant answer aa jaye. Trick ye hai ki poori table ratne ki zaroorat nahi: $7\times8 = 8\times7$ (order se farq nahi padta), aur ×9, ×5, ×10 jaise shortcuts se bade facts nikaale ja sakte hain. Ye "80/20" soch — thodi core cheezein yaad, baaki derive.

Bade numbers ke liye asli jaadu hai place value tod-do wala funda. $47 = 40+7$ . Toh $47\times23 = 47\times20 + 47\times3$ . Har chunk easy hota hai, phir sab add kar do. Yehi long multiplication hai. Jo chhota sa zero doosri line mein lagate ho, wo isliye ki 23 ka "2" actually 20 hai — units nahi, tens hai. Isliye uska product ek column left shift hota hai.

Area model isi cheez ka picture hai. Ek rectangle banao jiski width ek factor, height doosra. Dono sides ko $40+7$ aur $20+3$ mein todo — ban gaye 4 chhote boxes. Har box ka area ek partial product: $800, 140, 120, 21$ . Sab jodo → 1081. Long multiplication aur area model bilkul same cheez hain — bas ek numbers mein, ek picture mein. Isse "distributive law" ( $a(b+c)=ab+ac$ ) aankhon se dikh jaata hai, aur aage algebra mein $(x+a)(x+b)$ bhi isi tarah samajh aayega.