Times Tables

Cars come with 4 wheels, gloves have 5 fingers, and party bags hold 6 sweets each. Whenever things come in equal groups — packs, wheels, legs on spiders — knowing your times tables tells you the total in a flash, without counting every one.

A times table is just a list of multiples — the 2s, the 5s, the 10s, and so on. The 4 times table, for example, is

4,\ 8,\ 12,\ 16,\ 20,\ \dots

That is exactly skip counting by 4, written down once so you can remember it. Each number in the list is one more group of 4, so a table is the answers to 4\times 1,\ 4\times 2,\ 4\times 3,\ \dots all in a row. Press play: a marker hops along the number line, landing on each multiple and reading it aloud.

It really means equal groups

Behind every times-table fact is one simple picture: equal groups. 4 \times 3 means four groups of three. You don't have to know the answer by heart — you can always build it. Three ducks, then three more, then three more, then three more:

duck duck duck  |  duck duck duck  |  duck duck duck  |  duck duck duck

Count them all, or just skip-count the groups — 3,\ 6,\ 9,\ 12 — and you land on the answer:

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

So "times" is repeated adding: add the same amount once for every group. That is why the times table and skip counting give the very same list of numbers.

Suppose you have 2 baskets with 5 cats in each — that's 2 \times 5:

cat cat cat cat cat  |  cat cat cat cat cat

That's 5 + 5 = 10 cats. If instead you had 5 baskets with 2 cats each, you'd line them up differently — but you'd still count 10 cats. The amount doesn't care how you group it.

Now pick a number and skip-count it all the way to twelve times. Each step adds one more group — just like the hops above — and the last cell is n \times 12.

See it: an array of dots

Equal groups look tidiest when you line them up in rows and columns — that neat rectangle of dots is called an array. The number of rows times the number of columns is the total number of dots, so an array is a multiplication you can simply count. Press Refresh for a new one, and check it: count the rows, count the columns, multiply.

It sounds like a lot to learn, but two friends cut the work in half. First, order does not matter when you multiply:

a \times b = b \times a

This is called commutativity. Because 7\times 3 and 3\times 7 give the same answer, every fact you learn is really two facts — so you only have to learn about half of the whole table. (You can see it in the array: turn the rectangle of dots on its side and the rows become columns, but the dots don't change.)

Here's the whole table at once. Slide a row and a column — where they cross is \text{row} \times \text{column}. Now swap the two sliders: you land on a different square, but the same answer, because a \times b = b \times a. The table is a mirror image across its diagonal — that symmetry is commutativity.

Patterns that do the work for you

And second, some tables follow easy patterns you can spot in a moment:

Hold up all 10 fingers. To work out 9 \times 4, bend down your 4th finger (counting from the left). Now look: 3 fingers stand to its left and 6 fingers to its right — read them off as 36! It works all the way up to 9 \times 10: bend the 10th finger and you see 9 fingers, then 0 — 90. The fingers to the left and right always total 9, which is exactly the digit pattern above.

Learn the patterns, lean on commutativity, and the times tables shrink from a scary wall of numbers into a handful of friendly tricks.

Three worked examples

Each one uses the equal-groups idea, then a pattern to check the answer:

Two traps that catch everyone at first:

Khan Academy walks through the multiplication tables here: