Patterns and relationships

This means students will explore linear proportions in a variety of contexts. Linear proportions apply to situations which can be modelled using equivalent fractions, that is, a/b = c/d where a,b,c, and d are integers (usually whole numbers). So proportional reasoning pervades many of the outcomes in all three strands and includes many of the following contexts:

This means students will describe the function rule for a linear relationship as well as recognise recursive relationships where more complex relationships are involved. For example, given the pattern of fish made with matchsticks and counters below, students should be able to represent the relationships in a table and graph and use these representations to predict the terms in the sequence:

This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to look at several examples, notice what changes (variables) and what does not, use appropriate mathematical terminology and symbols to describe the pattern, and apply the generalisation to other examples. At Level Four students should be able to describe and apply the properties of multiplication and division as these operations apply to whole numbers. These properties include commutativity, distributivity, associativity, inverse and identity.

This means students will form and solve simple linear equations in the form y = mx + c, where x and y are related variables and where m is a whole number and c is an integer, for example q = 3p – c, or a + 5 = 4b. When the value of one variable is given the value of the other can be found by solving the equation, for example 3p – 6 = 18. Students should understand the equals sign as a statement of balance and know what operations to both sides of an equation preserve that balance, for example take off the same number from both sides.

This means students will recognise that a sequential pattern can be either spatial, for example .., or numeric, for example 1, 3, 5, 7... A pattern has consistency so further terms of it can be anticipated from those already known. The focus in this thread is that students become increasingly sophisticated at describing the relationships between variables found in sequential patterns.

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.

This means students will explore sequential patterns, either spatial, for example, , ... or numeric, for example, 1, 3, 5, 7 ... A pattern has consistency so further terms of it can be anticipated from those already known. In spatial patterns students should be able to identify the repeating element, for example,

Students at level two should understand that numbers are counts that can be split in ways that make the operations of addition, subtraction, multiplication and division easier. From Level One students understand that counting a set tells how many objects are in the set. Building on this thinking at Level Two is to realise that the count of a set can be partitioned and that the count of each subset tells how many objects are in that subset. Also required is understanding that partitions of a count can be recombined.

This means the students will explore sequential patterns. A sequential pattern is one in which further members of that pattern can be predicted from previous members. So ..., and 1, 3, 5, 7, ... are sequential patterns. At Level One students should be able to reproduce a given pattern using objects, drawings or symbols and continue the pattern on with justification.

This means students will understand the link between the cardinal and ordinal aspects of counting. The ordinal aspect refers to the fact that counting numbers have a conventional order. The last number in a count tells how many objects are in a set if all the objects are matched in one-to-one correspondence to the sequence of counting numbers. The next number in the counting sequence tells the result of adding an object while the number before in the sequence tells the count when an object is removed.