Number and Algebra: Patterns and relationships, Level 5
AO1: Generalise the properties of operations with fractional numbers and integers.
This means students will understand that to generalise means to establish properties that hold for all occurrences.
This involves the ability to examine a number of cases, define the variables involved, use appropriate mathematical terminology and symbols,
and ultimately reason with the properties themselves. Fractional numbers, for the purpose of this objective, are defined as rational numbers in the form a/b,
where a and b are whole numbers and b ? 0. At Level Five students should be able to demonstrate their understanding of the properties of addition, subtraction,
multiplication and division as these operations apply to fractional numbers and integers. These properties include commutativity, distributivity, associativity,
inverse and identity. Demonstration of understanding should involve applying these properties in solving a variety of problems, using the properties to solve
equations without calculating both sides, for example, 6 x 
= 3 x 70 + 3 x 9, justifying their responses to conjectures such as true/false statements, and expressing the generalisations algebraically, for example, the commutative property for multiplication of integers may be represented by a x b = b x a, where a and b are integers. Students should be able to express the operations on fractional numbers algebraically, for example, a/b + c/d = (ad + cb)/bd, and substitute number values into the equation to confirm that it holds for all addition examples they attempt.
AO2: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.
This means students will recognise the features of tabular, graph and equation representations of linear and quadratic relationships. This includes connecting constant first or second order difference in tables with linear and quadratic relations respectively, with the graph (linear and parabolic) and standard equation forms (y = mx + c and y = a x 2 + bx + c) for such relations. For example, given the spatial pattern below students can use a table, graph or equation to represent the relation and solve problems.
This includes finding both a recursive and direct (functional) rules and using them to find further terms using a spreadsheet or calculator, for example:
Students should also use these tabular and graphic representations for other relationships, such as simple exponential and step relations, but it is acceptable for them to use recursive rules for these more difficult relations.
Click to download a PDF of second-tier material relating to Level 5 Patterns and Relationships (527KB)
