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

1.1.4 · Maths › Arithmetic & Number Systems

Multiplication KYA hai?

$a \times b = b \times a (commutative)$

Tables 1–20 — woh 80/20 jo aapko bachata hai

Long multiplication — KYU kaam karta hai

Worked example

$47 \times 23$ (two-digit multiplier) $23 = 20 + 3$ split karo. Do partial products nikalo, phir jodo.

     4 7
   × 2 3
   ------
   1 4 1   ← 47 × 3   (7×3=21 write1 carry2; 4×3=12+2=14)
 9 4 0     ← 47 × 20  (pehle 0 likho, phir 47×2=94)
 ------
 1 0 8 1

Doosri row mein shift/zero kyun? 23 mein jo 2 hai woh 2 tens hai, isliye uska partial product $47 \times 20$ hai, jo 0 par khatam hota hai. Shift = 10 se multiply karna hai. Final: $47 \times 20 + 47 \times 3 = 940 + 141 = 1081$ .

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

Area model — sab kuch ke peeche ki picture

Worked example

$47 \times 23$ area model se Sides split karo: $47 = 40 + 7$ , $23 = 20 + 3$ . Char cells:

×	40	7
20	$800$	$140$
3	$120$	$21$

Sum $= 800 + 140 + 120 + 21 = 1081$ . Yeh long multiplication se kyun match karta hai: $800 + 140 = 940 = 47 \times 20$ (top row) aur $120 + 21 = 141 = 47 \times 3$ (bottom row). Same do partial products! Area model unhe bas aur zyada tod ke dikhata hai.

Common mistakes (Steel-manned)

Active recall

Recall 12 saal ke bachche ko explain karo (Feynman)

Bade numbers ko multiply karna floor tiling jaisa hai. Floor ko kuch saaf rectangles mein kato (tens aur ones), chhote rectangle mein tiles gino (easy!), phir sab jod do. Kabhi ek baar mein poora scary number multiply nahi karte — friendly chunks multiply karo aur jod do. "Long multiplication" yahi chunk-and-add hai, ek seedhe column mein likha hua, aur doosri line mein jo chhota zero lagate ho woh isliye hai kyunki woh digit actually tens tha, ones nahi.

Recall Forecast-then-verify

$34 \times 12$ compute karne se pehle: forecast karo ki yeh $34 \times 12 \approx 30 \times 12 = 360$ ke paas hai plus thoda aur → andaaz ~400. Ab verify karo: $34 \times 12 = 340 + 68 = 408$ . Forecast ke kaafi paas ✓ — accha sanity check ki kahin zero galat jagah toh nahi lagaya.

a \times b

ka repeated addition mein kya matlab hai?

a

copies of

b

jod do (=

b

copies of

a

a \times b = b \times a

kyun hai?

a

rows ×

b

columns ke dot grid mein same dots hain, chahe rows se count karo ya columns se (commutativity).

Distributive law batao.

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

n \times 9

ki trick?

n \times 10 - n

(jaise

7 \times 9 = 70 - 7 = 63

n \times 5

ki trick?

n \times 10

ka aadha (jaise

8 \times 5 = 80/2 = 40

47 \times 23

mein doosri partial-product row 0 se kyun shuru hoti hai?

2 matlab 2 tens hai, isliye woh row

47 \times 20

hai, jo 0 par khatam hoti hai.

Area model mein partial product kya hai?

Ek cell ki area = ek factor ke ek place-value chunk ka doosre factor ke ek chunk se multiplication.

47 \times 23

compute karo aur char cell areas batao.

800 + 140 + 120 + 21 = 1081

a (b + c) = ab + c

error ka fix?

a

har term ko multiply karta hai:

ab + a c

16 \times 15

splitting se?

16 \times 10 + 16 \times 5 = 160 + 80 = 240

Connections

Addition — carrying and place value (multiplication repeated addition tak reduce hoti hai)
Place Value & Number Systems (placeholder zero kyun kaam karta hai)
Distributive Law (har multiplication algorithm ka engine)
Division — inverse of multiplication
Squares & Square Roots (tables 1–20 inhe speed up karte hain)
Algebra — Expanding Brackets (area model → $(x + a) (x + b)$ )