Generalising patterns and relationships

Exploring patterns can lead to students generalising relationships.

Generalisation is noticing properties that consistently apply and sometimes defining the nature of those properties.

For example:

• when two odd numbers are added together, the answer is even
• when a number is divisible by 9, it is also divisible by 3.

The ability to recognise patterns and relationships is a key aspect to developing known facts by association.

The doubling and halving relationships between the twos, fours and eights basic multiplication facts build on known facts to establish new ones.

The properties of operations are important generalisations and include the:

• commutative property for addition and multiplication
• identity property for addition, subtraction, multiplication and division
• associative property for addition and multiplication
• distributive property for multiplication over addition.

When students understand the properties of operations they can transfer the principles to more complex problems (e.g. 9 \(\times\) 3 = 10 \(\times\) 3 – 3 so 99 \(\times\) 3 = 100 \(\times\) 3 – 3).

Recognising the inverse relationships between addition and subtraction, and multiplication and division, gives students greater flexibility in how they solve problems and check their answers. It also reduces the number of facts to be learned.