Multiplication

Suppose you have four plates, and each plate has three cookies. How many cookies altogether? You could add: 3 + 3 + 3 + 3 = 12. But adding the same number over and over is slow. Multiplication is the quick way to do exactly that — it adds equal groups in one step.

4 \times 3 = 3 + 3 + 3 + 3 = 12

We read the \times sign as "times", "lots of", or "groups of". So 4 \times 3 is four lots of three — four groups, three in each group. The answer, 12, is called the product.

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Four groups, three in each: 3+3+3+3, which is 4 \times 3 = 12. Count them and check! The dashed rings remind you the groups must each be the same size — that is what lets us multiply instead of add them one by one.

Here is the same idea as an animation. Press play to build a grid: each row is one equal group, and we count a whole row at a time instead of one by one — which is really just skip counting. Replay it for a fresh fact each time.

See it as an array (rows × columns)

When the groups are lined up neatly into rows and columns, the picture is called an array. The number of rows times the number of columns is the total number of counters. You can count each row (that is one group), and there are as many rows as there are groups. Press Refresh for a new array, then work out the product before peeking at the answer underneath.

See it as jumps on a number line

There is one more way to picture 4 \times 3: start at 0 and take four equal jumps, each three long. You land on 3, then 6, then 9, then 12:

0 \xrightarrow{+3} 3 \xrightarrow{+3} 6 \xrightarrow{+3} 9 \xrightarrow{+3} 12

Equal groups, an array, and equal jumps are three pictures of the very same idea: adding the same number again and again.

Order does not matter: 4 \times 3 = 3 \times 4

Tip an array on its side and the rows become columns — but the number of counters does not change. So four threes make the same total as three fours:

4 \times 3 = 12 = 3 \times 4

This handy rule is called the commutative law. It means you can always multiply in whichever order is easier — and it halves how many facts you really have to learn.


2 \times 5 = 10
5 \times 2 = 10

Two rows of five, or five rows of two — both are 10 counters. Same dots, just turned a quarter of the way round.

Worked examples

1. 5 \times 2 — five groups of two:

5 \times 2 = 2 + 2 + 2 + 2 + 2 = 10

2. A spider has 8 legs. How many legs on 3 spiders? That is three groups of eight:

3 \times 8 = 8 + 8 + 8 = 24

3. 6 \times 4 is awkward to add six times — so use commutativity and do 4 \times 6 instead (four sixes):

6 \times 4 = 4 \times 6 = 6 + 6 + 6 + 6 = 24 Two traps to dodge when you multiply:

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Five cows, and every cow has 4 legs — five equal groups of four. So the field holds 5 \times 4 = 20 legs, without crawling about counting one leg at a time. That is the whole point of multiplication: equal groups, counted fast.

Khan Academy introduces multiplication here: