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).
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 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 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 ÷ 2 = 5. This means 'divide 10 into 2 equal parts'.