NA3-7: Generalise the properties of addition and subtraction with whole numbers.

Elaboration on this Achievement Objective

This means students will generalise, which means to establish properties that hold for all instances. Generalisation begins with noticing patterns and relationships in a few specific instances, defining the variables involved, noticing the relationships between the variables, then using appropriate mathematical terminology and symbols to describe the relationships. At Level Three students develop many generalisations that allow them to perform mental strategies effectively. These generalisations include the commutative property of addition and multiplication, for example 7 x 8 = 8 x 7, the associative property of addition and multiplication, for example (2 x 3) x 4 = 2 x (3 x 4), the distributive property of multiplication, for example 8 x 7 = 8 x 5 + 8 x 2, the inverse relationships of addition and subtraction, and of multiplication and division, for example 6 x 7 = 42 so 42 ÷ 7 = 6, and identities for all four operations, for example 17 x 1 = 17, 17 ÷ 1 = 17. It is not expected that students use algebraic symbols to express these generalisations. However, students should be able to look for relationships across the equals sign in equations to determine missing numbers, for example 4 x 12 = box. x 6 without calculating 4 x 12.