Mathematics from zero
Multiplication
You have 4 bags, and each bag holds 3 apples. You could add 3 + 3 + 3 + 3 to find the total. But there is a shorter way to say “four threes”: four times three. That shortcut is multiplication.
After this lesson you can say what multiplication means, name its result, explain why the order of the two numbers does not matter, know what multiplying by 0 and by 1 do, and multiply a larger number by working a place at a time.
Multiplication is repeated addition of equal groups. When you have several groups
that are all the same size, multiplication finds the total fast. We write it with the
times sign, ×. 4 × 3 means “4 groups of 3”, which is 3 + 3 + 3 + 3 = 12. The
first number says how many groups, the second says how big each group is.
The numbers multiplied are factors; the result is the product. In 4 × 3 = 12,
the 4 and the 3 are the factors, and the 12 is the product. Multiplying
is just a quick way to reach a product you could also have reached by adding the same
number over and over.
You can reach the product by counting up in equal jumps. 4 × 3 is four jumps of
3 along the number line: 0 to 3 to 6 to 9 to 12. Each jump is one group. Land after
four jumps and you are on the product, 12.
The order of the factors does not matter. 4 × 3 and 3 × 4 both make 12.
Picture a rectangle of dots: 4 rows of 3 dots holds the same number of dots as 3 rows
of 4 — you just turned the rectangle on its side. So when one factor is small, put it
second and count that many jumps of the larger one: fewer jumps, same product.
Two factors run every multiplication: 0 and 1. Any number times 1 is itself —
one group of seven is just seven. Any number times 0 is 0 — zero groups of
anything hold nothing. And multiplying by 10 shifts every digit one place to the
left: 7 × 10 = 70, because each one becomes a ten. For larger numbers, split one
factor by place and multiply each part. 13 × 4 is (10 × 4) + (3 × 4).
Multiply 13 × 4.
Split 13 by place: it is 1 ten and 3 ones, so 13 = 10 + 3.
Multiply each part by 4. The tens part: 10 × 4 = 40. The ones part: 3 × 4 = 12.
Add the two partial products: 40 + 12 = 52.
So 13 × 4 = 52. Check the idea: 13 added four times is 13 + 13 + 13 + 13 — two
13s make 26, and two more make 52. Same product.
Why this works
Why can you split a factor by place and multiply the parts separately? Because multiplication shares out over addition: four groups of (10 + 3) is the same as four groups of 10 together with four groups of 3. You are not changing the total — only choosing an easier order to count it in. This is the idea every written multiplication method is built on.
Common mistake
A common mistake is multiplying only the ones digit and forgetting the tens — writing
13 × 4 = 12 because 3 × 4 = 12. Every part of the number must be multiplied. After
splitting by place, check that you used all the parts before adding the partial
products.
5 × 3 means five groups of three. Type the product.
Multiply: 4 × 0. Type the product.
Multiply: 6 × 1. Type the product.
7 × 8 makes 56. Without recounting, what does 8 × 7 make? Type the product.
Multiply: 13 × 5 by splitting 13 into 10 + 3. Type the product.
To multiply 13 × 4, you split 13 into 10 + 3. Why is that allowed?
Multiplication is repeated addition of equal groups: 4 × 3 is four groups of three.
The numbers multiplied are the factors; the result is the product. The order of the
factors never changes the product, so count jumps of the larger one. Multiplying by 1
leaves a number unchanged, by 0 gives 0, by 10 shifts each digit one place left. For
larger numbers, split a factor by place, multiply each part, and add the partial
products.