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.
- 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.
Commutative and associative properties
An understanding of the commutative property of addition and multiplication reduces the number of facts to be learned. The associative property of addition and multiplication assists when a calculation involves more than two numbers.
Distributive property of multiplication
The distributive property makes calculations which use larger numbers more accessible by splitting them into several easier mental calculations.
Recognising the inverse relationship between addition and subtraction, and multiplication and division, assists students to generate a family of facts for a given set of numbers.