# What is Multiplication?

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Your understanding of the key operations in math underpins your understanding of the whole subject. If you’re teaching young students or are just re-learning some elementary math, going over the basics can be very helpful. Most calculations you’ll need to do involve multiplication in some way, and the “repeated addition” definition really helps to cement what multiplying something means in your head. You can also think about the process in terms of areas. The multiplication property of equality also forms a core part of algebra, so it can be useful to go over at higher levels too. Multiplication really just describes calculating how many you end up with you have a specified amount of “groups” of a particular number. When you say 5 × 3, you are saying “What is the total amount contained within five groups of three?”

#### TL;DR (Too Long; Didn't Read)

Multiplication describes the process of repeatedly adding one number to itself. If you have 5 × 3, this is another way of saying “five groups of three,” or equivalently, “three groups of five.” So this means:

5 × 3 = 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 = 15

The multiplication property of equality states that multiplying both sides of an equation by the same number produces another valid equation.

## Multiplication as Repeated Addition

Multiplication fundamentally describes the process of repeated addition. One number can be considered the size of the “group,” and the other tells you how many groups there are. If there are five groups of three students, then you could find the total number of students using:

\text{Total number} = 3 + 3 + 3 + 3 + 3 = 15

You’d work it out like this if you just counted the students by hand. Multiplication is really just a shorthand way of writing out this process:

So:

\text{Total number} = 3 + 3 + 3 + 3 + 3 = 5 × 3 = 15

Teachers explaining the concept to third grade or elementary school students can use this approach to help cement the meaning of the concept. Of course, it doesn’t matter which number you call the “group size” and which one you call the “number of groups” because the result is the same. For example:

5 × 7 = 7 + 7 + 7 + 7 + 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35

## Multiplication and the Areas of Shapes

Multiplication is at the heart of the definitions for the areas of shapes. A rectangle has one shorter side and one longer side, and its area is the total amount of space it takes up. It has units of length2, for example, inch2, centimeter2, meter2 or foot2. No matter what the unit is, the process is the same. 1 unit of area describes a little square with sides 1 unit of length long.

For the rectangle, the short side takes up a certain amount of space, say 10 centimeters. This 10 centimeters repeats over and over as you move down the longer side of the rectangle. If the longer side measures 20 centimeters, the area is:

\begin{aligned} \text{Area} &= \text{width} × \text{length}\\ &= 10 \text{ cm} × 20 \text{ cm} = 200 \text{ cm}^2 \end{aligned}

For a square, the same calculation works, except the width and the length are really the same number. Multiplying the length of a side by itself (“squaring” it) gives you the area.

For other shapes, things get a bit more complicated, but they always involve this same key concept in some way.

## The Multiplication Property of Equality and Equations

The multiplication property of equality states that if you multiply both sides of an equation by the same quantity, then the equation still holds. So this means if:

a = b

Then

ac = bc

This can be used to solve algebra problems. Consider the equation:

\frac{x}{c} = \frac{12}{c}

This would be impossible to solve for ​x​ directly because you don’t know ​c​ either, but using the multiplicative property of equality, you can multiply both sides by ​c​ and write:

\frac{xc}{c} = \frac{12c}{c}

So

x = 12

Re-arranging equations works in a similar way. Imagine you have the equation:

\frac{x}{bc} = d

But want an expression for ​x​ alone. Multiplying both sides by ​bc​ accomplishes this:

\frac{xbc}{bc} = dbc \\ x=dbc

You can also use it to solve problems where you need to remove one quantity:

\frac{x}{3} = 9

Multiply both sides by three to get:

\frac{3x}{3} = 9×3 \\ x=27

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