Multiplication

6 plates of apples, 4 apples each — how many in total? Writing $4 + 4 + 4 + 4 + 4 + 4$ is tedious, so we invented a shortcut: $4 \times 6 = 24$ .

Multiplication is shorthand for adding the same number again and again. The 4 and the 6 are called factors, and the result 24 is the product.

Arrays: multiplication you can see

Lay out 24 dots in 4 rows of 6. Reading across gives "4 sixes"; reading down gives "6 fours" — the same array, read two ways. That is why $4 \times 6 = 6 \times 4$ : the commutative law is something you can see, not just memorize.

For speed, know your times tables well; combine them with the distributive law and even two-digit products become mental math.

InteractiveMultiplication Drill

6 × 4 = ?

🔥 0 · ★ 0

Three laws: swap, regroup, distribute

Commutative: $4 \times 6 = 6 \times 4$ — the same array viewed from another side;
Associative: in $2 \times 3 \times 5$ , computing $2 \times 5 = 10$ first and then multiplying by 3 is far easier than going in order;
Distributive: stuck on $7 \times 8$ ? Split it: $7 \times (5 + 3) = 7 \times 5 + 7 \times 3 = 35 + 21 = 56$ .

The distributive law is the most important of the three — long multiplication, mental math and, later, factoring all lean on it.

Long multiplication: split, then combine

To compute $24 \times 36$ , split 36 into 30 and 6, multiply 24 by each, and add:

$24 \times 36 = 24 \times 6 + 24 \times 30$

$24 \times 6 = 144$ ;
$24 \times 30 = 720$ (first $24 \times 3 = 72$ , then append a 0);
Combine: $144 + 720 = 864$ .

Common trap

The 0 in the second row, $720$ , must not be lost — it marks "30 times", not "3 times". Drop it and your answer shrinks by exactly ten times.

Mental math helper

$6 \times 15$ looks awkward? Split it: $6 \times 10 + 6 \times 5 = 60 + 30 = 90$ . Any number near a tidy ten will split this way.

Shortcuts for 10 and 100

$34 \times 10 = 340$ : every digit steps up one place, and a 0 is appended;
$34 \times 100 = 3400$ : two places up, two zeros appended.

The reason is place value again: multiplying by 10 moves each digit to a seat worth 10 times more. Run backwards, this shortcut is dividing by 10, and it is also the key to moving the decimal point — you will meet it again in decimals. Nail multiplication and division is half done.

Check yourself

Quick quiz

1. 24 × 36 = ?
2. By the distributive law: 7 × 5 + 7 × 3 = ?
3. 46 × 100 = ?