A pattern is a regularity or consistency in the arrangement of elements, such as in numbers, letters, shapes, objects or colours.

In early algebra, there are two main types of patterns:

• repeating patterns

• growing patterns.

A repeating pattern has a unit of repeat.

A growing pattern increases or decreases in a consistent manner. Number patterns are a type of growing pattern.

In mathematics, patterns are important because they can usually be expressed as generalisations. In the pattern shown in Figure 2, students might find a rule for progression of the pattern, such as: ‘The llama grows by 4 tiles each year’. They might also find a direct rule for any year, such as: ‘To find the number of tiles, multiply the year by 4 and add 6'.

Patterns can also be found in the behaviour of numbers when they are added, subtracted, multiplied or divided.

A variable is a letter or symbol used in algebraic expressions and equations. It is a letter that can have different values in the same problem. Letters are used most powerfully to represent relationships between variables (quantities that change).

A function is a relationship that exists between variables. Each value of the input (independent variable) maps to only one value of the output (dependent variable). For example, the relationship between the number 7 and the number 5 is minus 2.

In the function Δ + 5 = ◻, each value of Δ results in one value of ◻.

Specific inputs and outputs can be written as ordered pairs.

An expression is a mathematical phrase with two or more terms (numbers or letters) connected by operations. Expressions do not have an equals sign.

An equation is a mathematical statement with two or more expressions that are equal in value. An equation must have an equals sign.

Number properties make an expression easier to work with and are applied when students perform calculations.

The most useful laws are the commutative, distributive and associative laws of addition and multiplication, as well as the inverse relationships between addition and subtraction, and between multiplication and division.

A student who solves 3 + 99 using 99 + 3 applies the commutative law of whole numbers under addition.

The following example uses the distributive and associative properties of whole numbers under addition.

A generalisation is a statement that holds true in all cases or for a specified set of cases. The process is: seeing features in examples, noticing a pattern, forming a conjecture about the pattern, justifying the conjecture using structure and generalising the rule.

Relational thinking is the manipulation of both sides of an equality without needing closure. Equality is maintained if the same operation is applied to both sides of an equation.

In the above instance, one is added to each addend.

Relational thinking involves applying relationships within equations, across the equality, to find unknowns.