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'.
Division, just like the other operations, has specific language for each part of an equation. For example:
We might have heard of dividend from the world of finance, meaning a share of profits. Here, it is the value we are dividing. The divisor tells us how many times we must divide by and the answer to a division is known as the quotient. This has etymology in latin: quotiens meaning 'how many times'.
Division, just like subtraction, does not follow the commutative law so: 10 ÷ 5 ≠ 5 ÷ 10.
This method is sometimes known as the 'bus stop' method because it vaguely resembles a bus stop. Such tricks are just that, a trick which deflects away from the concept of why and how the short division method works.
Notice that we are now working from left to right, not right to left as we have with addition, subtraction and multiplication. We might reasonably ask: 'how many groups of 5 go into 6' the answer is 1 with 1 remaining, which is carried over to the next digit. But we should consider what this means mathematically rather than trying to memorise a process which does not reveal mathematical structure.
Here, we are showing that is 1 group of one hundred 5s in 600, leaving a remainder of 100. In the next column, we have shown that there are 3 groups of ten 5s in 150 and finally we show that there is 1 group 5 in 5.
This is an area of mathematically that ca be notoriously difficult to teach. This may be for many reasons, as we will see, to do with times tables knowledge, memorising unrelated, abstract processes and wondering why we are adding when we are meant to be dividing.
Let's take the example 768 ÷ 24. Already, children may look at this and attempt to to use the short division method, but as the divisor is unfamiliar, this can leave a lot of room for error. Here, I offer a method which can support children. Often, children struggle in being accurate with the multiples of the divisor. Partitioning the divisor and looking for patterns can be an effective strategy as can using multiple tracks. Both require looking for patterns and links.
After the multiples are accurate, it is time to begin chunking.
One this is complete, you may want to encourage children to look for patterns such as:
768 ÷ 24 = 32
769 ÷ 24 = 32 r 1
770 ÷ 24 = 32 r 2
771 ÷ 24 = 32 r 3
...and so on.
You might also generalise that the divisor can never be greater than the remainder, so when we arrive at remainder 23, we then have an exact fit for 792 ÷ 24 = 33.
We may want to simplify this concept first to:
10 ÷ 5 = 2
11 ÷ 5 = 2 r 1
12 ÷ 5 = 2 r 2
13 ÷ 5 = 2 r 3
14 ÷ 5 = 2 r 4
15 ÷ 5 = 3
16 ÷ 5 = 3 r 1
17 ÷ 5 = 3 r 2
18 ÷ 5 = 3 r 3
19 ÷ 5 = 3 r 4
20 ÷ 5 = 4