• knowledge of the effect of multiplying by ten on the location of the decimal point in a number
• use of lists, venn diagrams and grids to record items that have a certain attribute
• selection of a sample from a population
• recognition that one set is or is not a subset of another
• use of ‘=’ to indicate equivalence or the result of a computation

• The meaning of the equals sign

• variation of order and grouping of addition (commutative and associative property) to facilitate computations; for example, 
  3 + 5 + 7 + 5
  = 3 + 7 + 5 + 5
  = 10 +10 = 20
• specification of all possible outcomes of a simple chance event
• construction of number sentences
• calculations using notation such as ‘3 + 5 − 2 =’

• Construction of Number Sentences

At 2.75, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• use of distributive property in calculations; for example, 6 × 37
  = 6 × 30 + 6 × 7
• construction of lists, venn diagrams and grids to be used for recording combinations of two attributes

• Properties of Operations

At Level 3, students recognise that the sharing of a collection into equal-sized parts (division) frequently leaves a remainder. They investigate sequences of decimal numbers generated using multiplication or division by 10. They understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence). They use number properties in combination to facilitate computations (for example, 7 + 10 + 13 = 10 + 7 + 13 = 10 + 20). They multiply using the distributive property of multiplication over addition (for example, 13 × 5 = (10 + 3) × 5 = 10 × 5 + 3 × 5). They list all possible outcomes of a simple chance event. They use lists, Venn diagrams and grids to show the possible combinations of two attributes. They recognise samples as subsets of the population under consideration (for example, pets owned by class members as a subset of pets owned by all children). They construct number sentences with missing numbers and solve them.

• conversion between venn diagrams and karnaugh maps as representations of relationships between two sets
• recognition and completion of patterns formed by constant addition or subtraction
• use of add and subtract as inverse operations to solve simple word equations such as ‘I am thinking of a number. If I add 6 I get 18, what number did I start with?’
• use of trial and error to find a missing number in a number sentence; for example, 4 × ? + 6 = 22
• use of language to describe change in everyday items or attributes whose value varies over time

• Missing Number Sentences

At 3.5, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• incorporation of tables of information relating pairs of everyday variables
• sorting of sequences into certain types (constant addition, constant multiplication, fibonacci, square, triangular)
• use of division and multiplication as inverses; for example, multiplication by 25 can be carried out as ‘multiplication by 100 followed by division by 4’
• consistent and correct use of conventions for order of operations

At 3.75, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• construction of diagrams illustrating the possible relationship between two sets and the truth of statements involving the words all, some or none
• construction of number patterns and tables of values from an equation or a recurrence relation
• recognition that a given number pattern can be represented by an apparently unrelated equation and recurrence relation; for example, 5, 9, 13 … represented by ‘multiply position in the pattern (first, second, third ...) by 4 and add 1’ and ‘start with 5 then repeatedly add 4 to the previous term’
• understanding of zero and its characteristic of not having a multiplicative inverse, and the consequences of attempting division by zero

• Venn diagrams

At Level 4, students form and specify sets of numbers, shapes and objects according to given criteria and conditions (for example, 6, 12, 18, 24 are the even numbers less than 30 that are also multiples of three). They use venn diagrams and Karnaugh maps to test the validity of statements using the words none, some or all (for example, test the statement ‘all the multiples of 3, less than 30, are even numbers’).

Students construct and use rules for sequences based on the previous term, recursion (for example, the next term is three times the last term plus two), and by formula (for example, a term is three times its position in the sequence plus two).

Students establish equivalence relationships between mathematical expressions using properties such as the distributive property for multiplication over addition (for example, 3 × 26 = 3 × (20 + 6)).

Students identify relationships between variables and describe them with language and words (for example, how hunger varies with time of the day).

Students recognise that addition and subtraction, and multiplication and division are inverse operations. They use words and symbols to form simple equations. They solve equations by trial and error.

• Rules for Sequences
• Equivalence in number sentences

At 4.25, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• use of inverse and identity when subtracting and dividing rational numbers
• identification of domain and range; independent and dependent variable and their role in graphing
• representation of data by plotting points in the first quadrant and explanation of key features
• collection and classification of sets of data as either linear or non-linear depending on whether the slope is constant
• interpretation of a letter as a symbol for any one of a set of numbers and use in symbolic description of relationships

• The meaning of letters in algebra

At 4.5, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• use of inequality, equality, approximately equal and not equal, including in symbolic expressions
• translation from verbal description to algebraic representation, and of the structure of algebraic expressions; for example, if $500 is shared between n people, each receives ⁵⁰⁰/_n
• solution of simple linear equations using tables, graphs and inverse operations (backtracking)
• representation of inequalities as parts of the number line; for example, x < −5
• translation between symbolic rules, patterns and tables for linear functions

• Structure of algebraic expressions

At 4.75, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• lists of sets in the power set of a given set and knowledge that the total number of set equals 2ⁿ for n elements in the given set
• solution of equations such as x² = 17 as required in measurement situations; for example, using pythagoras theorem
• graphical representation of simple inequalities such as y ≤ 2x + 4
• selection of a type of function (linear, exponential, quadratic) to match a set of data
• translation between forms (table, graph, rule, recurrence relation) of representation of a function

At Level 5 students identify collections of numbers as subsets of natural numbers, integers, rational numbers and real numbers. They use Venn diagrams and tree diagrams to show the relationships of intersection, union, inclusion (subset) and complement between the sets. They list the elements of the set of all subsets (power set) of a given finite set and comprehend the partial-order relationship between these subsets with respect to inclusion (for example, given the set {a, b, c} the corresponding power set is {Ø, {a}, {b}, {c }, {a, b }, {b, c}, {a, c }, {a, b , c}}.)

They test the validity of statements formed by the use of the connectives and, or, not, and the quantifiers none, some and all , (for example, ‘some natural numbers can be expressed as the sum of two squares’). They apply these to the specification of sets defined in terms of one or two attributes, and to searches in data-bases.

Students apply the commutative, associative, and distributive properties in mental and written computation (for example, 24 × 60 can be calculated as 20 × 60 + 4 × 60 or as 12 × 12 × 10). They use exponent laws for multiplication and division of power terms (for example 2³ × 2⁵ = 2 ⁸, 2⁰ = 1, 2³ ÷ 2⁵ = 2^-2 , (5²)³ = 5⁶ and (3 × 4)² = 3 ² × 4²).

Students generalise from perfect square and difference of two square number patterns
(for example, 25² = (20 + 5)² = 400 + 2 × (100) + 25 = 625. And 35 × 25 = (30 + 5) (30 - 5) = 900 - 25 = 875)

Students recognise and apply simple geometric transformations of the plane such as translation, reflection, rotation and dilation and combinations of the above, including their inverses.

They identify the identity element and inverse of rational numbers for the operations of addition and multiplication
(for example, ½+ - ½ = 0 and ² / ₃ × ³ /₂ = 1).

Students use inverses to rearrange simple mensuration formulas, and to find equivalent algebraic expressions
(for example, if P = 2L + 2W, then W = ^P / ₂ - L . If A = p r² then r = v ^A /p).

They solve simple equations (for example, 5 x + 7 = 23, 1.4x - 1.6 = 8.3, and 4x ² - 3 = 13) using tables, graphs and inverse operations. They recognise and use inequality symbols. They solve simple inequalities such as y = 2x+ 4 and decide whether inequalities such as x ² > 2y are satisfied or not for specific values of x and y.

Students identify a function as a one-to-one correspondence or a many-to-one correspondence between two sets. They represent a function by a table of values, a graph, and by a rule. They describe and specify the independent variable of a function and its domain, and the dependent variable and its range. They construct tables of values and graphs for linear functions. They use linear and other functions such as f( x) = 2x - 4, xy = 24, y = 2^x and y = x² - 3 to model various situations.

• Sets
• Manipulating symbols

At 5.25, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• relationships between two sets using a venn diagram, tree diagram and karnaugh map
• factorisation of algebraic expressions by extracting a common factor
• solution of equations by graphical methods
• identification of linear, quadratic and exponential functions by table, rule and graph in the first quadrant
• knowledge of the quantities represented by the constants m and c in the equation y = mx + c

• Conceptual growth for solving equations

At 5.5, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• expression of the relationship between sets using membership, ∈, complement, ′ , intersection, ∩, union, ∪, and subset, ⊂, for up to two sets
• representation of numbers in a geometric sequence (constant multiple, constant percentage change) as an exponential function
• knowledge of the relationship between geometrical and algebraic forms for transformations
• expansion of products of algebraic factors, for example, (2x + 1)(x − 5) = 2x² − 9x − 5
• equivalence between algebraic forms; for example, polynomial, factorised and turning point form of quadratics
• use of inverse operations to re-arrange formulas to change the subject of a formula

• Exponential functions

At 5.75, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• expression of irrational numbers in both exact and approximate form
• factorisation of simple quadratic expressions and use of the null factor law for solution of equations
• testing of sequences by calculating first difference, second difference or ratio between consecutive terms to determine existence of linear, quadratic and exponential functions
• formulation of pairs of simultaneous equations and their graphical solution
• representation of algebraic models for sets of data using technology

At Level 6, students classify and describe the properties of the real number system and the subsets of rational and irrational numbers. They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations.

Student express relations between sets using membership, ?, complement, ', intersection, n, union, ?, and subset, ?, for up to three sets. They represent a universal set as the disjoint union of intersections of up to three sets and their complements, and illustrate this using a tree diagram, venn diagram or karnaugh map.

Students form and test mathematical conjectures; for example, ‘What relationship holds between the lengths of the three sides of a triangle?’

They use irrational numbers such as, p, f and common surds in calculations in both exact and approximate form.

Students apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to computation with number, to rearrange formulas, rearrange and simplify algebraic expressions involving real variables. They verify the equivalence or otherwise of algebraic expressions (linear, square, cube, exponent, and reciprocal,
(for example, 4 x - 8 = 2(2x - 4) = 4(x - 2); (2a - 3)² = 4a² - 12a + 9; (3w)³ = 27w³ ; ( /x = x²y ^-1; 4 / x y = 2/ x × 2/ y).

Students identify and represent linear, quadratic and exponential functions by table, rule and graph (all four quadrants of the Cartesian coordinate system) with consideration of independent and dependent variables, domain and range. They distinguish between these types of functions by testing for constant first difference, constant second difference or constant ratio between consecutive terms (for example, to distinguish between the functions described by the sets of ordered pairs
{(1, 2), (2, 4), (3, 6), (4, 8) …}; {(1, 2), (2, 4), (3, 8), (4, 14) …}; and {(1, 2), (2, 4), (3, 8), (4, 16) …}). They use and interpret the functions in modelling a range of contexts.

They recognise and explain the roles of the relevant constants in the relationships f(x) = a x + c, with reference to gradient and y axis intercept, f(x) = a (x + b)² + c and f(x ) = ca^x.

They solve equations of the form f(x) = k, where k is a real constant (for example, x(x + 5) = 100) and simultaneous linear equations in two variables (for example, {2x - 3y = -4 and 5x + 6y = 27} using algebraic, numerical (systematic guess, check and refine or bisection) and graphical methods.