Maths is full of symbols. Some of the more familiar are that addition and subtraction symbols. The equals symbol is a relative newcomer having been introduced in 1557 by Welsh mathematician Robert Recorde.

__Addition__

When we add, we sum together two (or more) quantities. This results in an increase in quantity (__unless we are working with negative numbers, of course__). Numbers being added together have useful names which can aid children to reason clearly. Let's take 9 + 1 = 10. 9 is the **augend**: the number in which is to be added to. 1 is the **addend**: the number too add to the **augend**. The total is 10 and so the **sum** is 10. As children gain fluency is adding 1, they begin to make links and connections known as **number bonds** which make patterns and can expose the structure of the number system.

This may be the first time children look for structures and make generalisations in maths. They are useful to know as their recall aids fluency with arithmetic and understanding of the number system which is built around sets of 10 (including the decimal system).

__Laws__

The three fundamental laws of addition unlock the structure of maths: **associative** law, **commutative** law and the **distributive** law. Understanding how the laws work leads to a deep understanding of how mathematics works.

__Associative law.__

This law states it does not matter how we group numbers with addition for the **sum** will always be the same. This law allows children to make calculations more efficient. For example:

9 + 12 + 8: it is more efficient to do 9 + (12 + 8) than (9 + 12) + 8. **Number bonds** to 20 help here too.

__Commutative law__

The commutative law states the order of addition makes no difference to the **sum**. For example: 1 + 2 = 2 + 1.

This is useful as children may encounter problems such as: 9 + 12 + 8 + 1. It would be much more efficient to look for number bonds such as 9 + 1 + 12 + 8 and achieving the same **sum**. We can express this algebraically: *a + b = b + a*.

__Distributive law__

This law requires numbers to be 'broken down' in small components. This is an essential skill for children to master to look for numbers *within* numbers. Essentially, looking for **number bonds** within a set of numbers. Let's look at this example:

27 + 14 + 11.

Here, we want to encourage children to apply the **distributive law** to break the problem down. It may look something like this:

(20 + 7) + (10 + 4) + (10 + 1)

(20 + 10+ 10) + (7 + 4 + 1)

(40 + 12)

= 52

__Column methods__

__Addition__

The idea of breaking numbers down into smaller components (the distributive law) is an essential concept to mastery for mathematical success. Here, the column method of addition breaks (**partitions**) a number into its **place value**. Many teachers, parents and children will be familiar with this method.

This method works by essentially transforming a calculation into (6 + 3) + (50 + 20) + (400 + 100). When we cross the tens boundary, we must **carry** to the greater place value section. Without carrying, calculations would look like this:

Using place value counters can be an effective way to demonstrate how the column method works efficiently. The highlight ones in the sum, show that **10 ones are exchanged for 1 ten.**

__Subtraction__

Subtraction is the concept of reducing a number or quantity by another. It 's etymology lies in the latin for 'removal'.

Take the example: 5 - 2 = 3.

We can represent this in many different ways.

__Laws__

Unlike addition, subtraction is not commutative and must be done in order (there are some exceptions to subtraction calculation with more than 2 values: 25 - 5 - 3 - 2) (This is because the subtrahends can be added together and addition is **commutative**).

Therefore 5 - 3 **≠ **3 - 5.

As we saw with **augend**, **addend** and **sum** in addition, subtraction too has specific names for each part of a calculation. There is a good reason for this as it avoids ambiguity and confusion when reasoning.

__Associative__

Subtraction is not associative because: (12 - 5) - 2 **≠ **12 - (5 - 2). This means we would be changing the value of the **subtrahend**.

__Same difference__

Understanding that subtraction can be considered as finding the** difference** between two numbers and changing the **subtrahend** and** minuend** by the same amount, the **difference** will always remain the same.

For example: 10 - 8 = 2, 6 - 4 = 2, 5 - 3 = 2 and 3 - 1 = 2. The difference between the **subtrahend** and **minuend** are always 2. This can be useful for calculations such as: 33 - 7, changing the **subtrahend** and **minuend** so they have the same difference (36 - 10 = 26) allows for efficient use of this strategy.

__Column subtraction__

Column subtraction is visually similar to column addition. We place each digit in its place value position and stack them up.

However, the method needs a different approach when the digit in the **subtrahend** is greater than the corresponding digit in the **minuend**. I am sure many children revert to looking for the difference between two digits and placing this in the answer. In the example below, we can almost make sense of not exchanging (300 and 30 subtract 1 = 329). This however, does not use the decimal form.

Referring to each digit by its place value can support understanding here. Exchanging 1 ten from the 5 tens and placing the exchanged ten in the ones column creates 16 - 7.

This can be avoided all together using the same difference concept. Using place value counters, in a similar way as we did this addition, can strengthen and deepen understanding of the column method of subtraction.

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