# How I wish I had always taught... multiplication and division.

Most teachers know that multiplication can be modeled as repeated addition. Although this does not cover more complex examples of multiplication, it is a sound starting point to make sense of multiplication. For example 3 x 5 = 3 x 3 x 3 x 3 x 3.

Representation and visualisation are important here when making the connection between repeated addition and multiplication. Focusing on the vocabulary of multiplication can help us reason the correct way of representing 3 x 5. The first number is referred to as the **multiplier** (3), and the second number is referred to as the **multiplicand**. Therefore, technically 3 should be represented five times, rather than 5 represented 3 times. However, as we will see, multiplication is commutative and therefore is may be most useful to refer to the numbers in the calculation as **factors**, which when multiplied together, create a **product** (the answer).

__Laws__

The laws of multiplication are exactly the same as the laws of __addition__. Multiplication is **commutative** so:

**3 x 5 = 5 x 3** and so **a x b = b x a**

Multiplication is also ** associative** so (3 x 5) x 7 = 3 x (5 x 7) and so (

**a x b) x c**=

**a x (b x c)**.

Also, multiplication (and addition) are **distributive**. So, multiplication distributes over addition.

**3 (5+7) = 3 x 5 + 3 x 7** and so **a (b+c) = ab + ac.**

Using the **distributive** law in multiplication helps expose the structure of multiplication. For example, 38 x 7.

**38 x 7 = 30 x 7 + 8 x 7**, it could even be made easier by further distribution: **37 x 7 + 8 x 7 =10 x 7 + 10 x 7 + 10 x 7 + 8 x 7**. Manipulating multiplication in this way is important for students to master so they may confidently, fluently and accurately calculate and reason.

__Column multiplication__

Column multiplication is visually similar to that of __column addition__. The columns denote the value of each digit. Here, we apply the law of distribution by effectively doing **654 x 2 = 2 x 4 + 2 x 50 + 2 x 600 **as we consider the value of each digit.

Long multiplication is slightly more complex but it based around manipulating the laws of multiplication. Let's look at 654 x 23.

Here we essentially calculate: 3 x 654 + 20 x 654 and then combine both **products. **When this becomes clear to children that the value of the digit '2' in this calculation has a value of 20, we avoid needed to 'add a zero' as we are actually multiplying the number by ten as the '2' is in the tens place value column.

__Division__

Division is the **inverse** of multiplication. There are several ways division may be represented but we tend to use the division symbol (the obelus) or were write it as a fraction. We might show 10 รท 2 = 5. This means '**divide 10 into 2 equal parts**'.