Multiplication
Multiplication is the shortcut for adding the same amount again and again. Six groups of four, a rectangle six rows deep, or four added six times — all three are the same twenty-four.
The multiplication tool
Build the array
Drag the corner handle
6 4 24
6 rows of 4.
How multiplication works
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See the equal groups
The multiplication sign says "groups of this size". Six times four means six groups, each holding four, and the answer is how many in total. Writing it as four added six times gives the same amount, which is why multiplication is described as repeated addition — the pattern you already met in addition, with every group the same size.
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Draw it as a rectangle
Put the groups in rows and you have an array: six rows of four dots. Count the squares, not the lines — a rectangle drawn with 7 vertical lines has only 6 columns. Then count in a way that suits the number: six rows of four is the same as counting 4, 8, 12, 16, 20, 24.
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Turn it sideways
Rotate the array a quarter turn and the rows become columns: six rows of four becomes four rows of six. Nothing was added or removed, so the total is unchanged. That is why the order of the two numbers never matters in multiplication, and it halves the number of facts you need to learn.
Worked examples
6 × 4
Six groups of four, six rows of four, or four added six times. Counting the array in fours gives 24. Turning it sideways shows 4 × 6 as the same array, so the answer is 24 there too.
7 × 8
Split the seven into five and two: 8 × 5 = 40 and 8 × 2 = 16, and 40 + 16 = 56. Splitting one factor into two easier parts and adding the pieces works for every fact you have forgotten, and it explains why the tables fit together.
9 × 7
Think of ten sevens first: 10 × 7 = 70. Nine groups is one group fewer, so take one seven away: 70 − 7 = 63. The nines table is the tens table minus one more row of the other factor.
Where it goes wrong
- Counting the lines instead of the squares An array drawn with 7 vertical and 6 horizontal lines has 6 columns and 5 rows, so the product is 30, not 42. The lines are the boundaries; the squares between them are the things being counted.
- Reading 3 × 4 as 3 + 4 The two signs ask different questions. 3 + 4 adds two different amounts and gives 7. 3 × 4 is four groups of 3, which is 3 + 3 + 3 + 3 = 12. Read the sign out loud as "groups of" until the difference sticks.
- Assuming a bigger factor means a bigger product A larger number does not automatically win: 9 × 2 = 18 is less than 5 × 5 = 25. Both factors decide the product, so check a surprising answer by picturing the array or splitting it into parts you know.
Questions
What does multiplication actually mean?
It means equal groups. The first number tells you how many groups, the second tells you how many are in each one, and the product is the total. Every multiplication can also be read as an array, as repeated addition, and as a rectangle whose rows and columns are the two factors. Being able to move between those readings is what makes the tables make sense.
Why is 6 × 4 the same as 4 × 6?
Draw six rows of four dots, then turn the paper a quarter turn. You are now looking at four rows of six, and no dots were added or taken away. Multiplication asks for a total, and the total does not care which way you face the rectangle. This is called the commutative law, and it means that learning one fact gives you another for free.
Which times tables should we learn first?
Start with the tens and the fives, because counting in tens and fives is familiar, then the twos, because they are doubles. Next come the squares such as 4 × 4 and 7 × 7, and the facts that turn up in pairs through the commutative law. The sixes, sevens and eights are left until last, once there are more known facts to lean on.
Is memorizing the times tables worth it?
Worth it, but after understanding, not instead of it. Recalling 7 × 8 instantly leaves your attention free to work on the harder step in a problem, while counting up in sevens has to be redone every time. Build the meaning first with arrays and repeated addition, then practise recall, and keep a strategy such as 10 × 8 − 8 for the facts that will not settle.
How does multiplication connect to division?
Every multiplication fact is also two division facts: from 6 × 4 = 24 you can say 24 ÷ 6 = 4 and 24 ÷ 4 = 6. Division asks which factor is missing, so a secure times table is what makes division quick rather than a search. That is also why division is the last of the four operations to be taught.