0.5

At 0.5, the work of a student progressing towards the standard at Level 1 demonstrates, for example:

• selection of appropriate materials for illustrating a mathematical problem or its solution
• use of drawing to represent problems and solutions
• verbal description of simple patterns and extension of these patterns
• recognition and use of numbers on a calculator
• recognition that addition is relevant to a task

• Simple Patterns

1.0 Standard

At Level 1, students use diagrams and materials to investigate mathematical and real life situations. They explore patterns in number and space by manipulating objects according to simple rules (for example, turning letters to make patterns like bqbqbq, or flipping to make bdbdbdbd).

They test simple conjectures such as ‘nine is four more than five’. They make rough estimates and check their work with respect to computations and constructions in Number, Space, and Measurement, chance and data. They devise and follow ways of recording computations using the digit keys and +, - and = keys on a four function calculator.

They use drawing tools such as simple shape templates and geometry software to draw points, lines, shapes and simple patterns. They copy a picture of a simple composite shape such as a child’s sketch of a house.

• Making better estimates

1.25

At 1.25, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• development of descriptive rules for patterns
• use and justification of approximations
• elementary use of mathematical symbols to describe their own thought processes
• reading and rewriting of numbers from a calculator
• checking of calculations using technology

1.5

At 1.5, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• formulation and testing of conjectures using models that involve, for example, objects, patterns, shapes and numbers
• verification of estimation of a solution to a number sentence
• understanding of how to follow a sequence of steps in a procedure
• addition of numbers on a calculator and recognition of the function of calculator keys
• understanding of appropriate action for responding to an incorrect calculator result

• Using a calculator

1.75

At 1.75, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• continuation of patterns and the recognition of inconsistencies
• search for alternative methods in order to verify answers
• assessment of the context at hand, and an explanation of this assessment
• representation of data using pictographs that are either hand-drawn or assisted by technology

• Recognising and using patterns

2.0 Standard

At Level 2, students make and test simple conjectures by finding examples, counter-examples and special cases and informally decide whether a conjecture is likely to be true. They use place value to enter and read displayed numbers on a calculator. They use a four-function calculator, including use of the constant addition function and x key, to check the accuracy of mental and written estimations and approximations and solutions to simple number sentences and equations.

2.25

At 2.25, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• identification of pattern and similarity in data sets and shapes, and use of pattern, often by observing a set of simpler situations, as a problem solving strategy
• use of materials and models to solve problems and explain answers
• checking of accuracy of calculations with a calculator
• use of technology to create and manipulate shapes and simple maps

2.5

At 2.5, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• selection of appropriate situations for the use of a guess–check–improve strategy
• explanation and comparison of alternative computation methods

• Guess-Check-Improve strategy

2.75

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

• use of materials and models to illustrate and test generalisations
• rephrasing of a problem or representing it using a physical model, diagram, list or table as a problem solving strategy
• selection of multiplication and division as more efficient processes than repeated addition and subtraction
• application of number skills to solve routine problems from everyday contexts
• partitioning of a task into smaller sub-tasks

• Using diagrams and models

3.0 Standard

At Level 3, students apply number skills to everyday contexts such as shopping, with appropriate rounding to the nearest five cents. They recognise the mathematical structure of problems and use appropriate strategies (for example, recognition of sameness, difference and repetition) to find solutions.

Students test the truth of mathematical statements and generalisations. For example, in:

• number (which shapes can be easily used to show fractions)
• computations (whether products will be odd or even, the patterns of remainders from division)
• number patterns (the patterns of ones digits of multiples, terminating or repeating decimals resulting from division)
• shape properties (which shapes have symmetry, which solids can be stacked)
• transformations (the effects of slides, reflections and turns on a shape)
• measurement (the relationship between size and capacity of a container).

Students use calculators to explore number patterns and check the accuracy of estimations. They use a variety of computer software to create diagrams, shapes, tessellations and to organise and present data.

3.25

At 3.25, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• consideration of problems with a similar mathematical structure as a problem solving strategy
• use of familiar problems to focus on strategies to help in solving an unfamiliar problem
• search for counter-examples in an attempt to disprove a conjecture
• location of data sources, including use of the world wide web
• collection of mathematical data using technology; for example, using data logging

3.5

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

• application of mathematics to model and solve simple practical problems; for example, the construction of a pair of stilts
• efficient communication when using mathematical language, symbols and representations
• appreciation of the history of mathematics in development of geometry and number concepts
• development and testing of conjectures with the aid of a calculator; for example, divisibility tests
• incorporation of text, data, images and graphs using technology, to report the results of an investigation

• Explain how maths is useful

3.75

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

• knowledge of interpretation of maps, graphs and models
• understanding of patterns through the use of systematic strategies such as calculating first differences
• application of a set of questions linked to an area of investigation
• knowledge of appropriate historical information

4.0 Standard

At Level 4, use students recognise and investigate the use of mathematics in real (for example, determination of test results as a percentage) and historical situations (for example, the emergence of negative numbers).

Students develop and test conjectures. They understand that a few successful examples are not sufficient proof and recognise that a single counter-example is sufficient to invalidate a conjecture. For example, in:

• number (all numbers can be shown as a rectangular array)
• computations (multiplication leads to a larger number)
• number patterns ( the next number in the sequence 2, 4, 6 … must be 8)
• shape properties (all parallelograms are rectangles)
• chance (a six is harder to roll on die than a one).

Students use the mathematical structure of problems to choose strategies for solutions. They explain their reasoning and procedures and interpret solutions. They create new problems based on familiar problem structures.

Students engage in investigations involving mathematical modelling. They use calculators and computers to investigate and implement algorithms (for example, for finding the lowest common multiple of two numbers), explore number facts and puzzles, generate simulations (for example, the gender of children in a family of four children), and transform shapes and solids.

• Counter-examples
• Real world investigations

4.25

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

• consideration of evidence to support theorems; for example, in geometry
• exploration of the appropriateness of linear models for data
• translation between verbal descriptions and algebraic rules
• use of technology to extend their own ability to make and test conjectures
• use of spreadsheets to manipulate data and generate graphs

4.5

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

• application of logic to the creation and use of a database
• identification of the mathematical information needed to solve a problem or carry out an investigation
• development of deductive proof to reach new conclusions
• use of interpolation to make predictions
• development of simple geometric and algebraic models for real situations; for example, representation of an animal as a cylinder

• Carrying out investigations

4.75

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

• communication of the results of a mathematical investigation in an appropriate form
• creation and manipulation of tables and graphs using technology
• numerical and graphical solution of algebraic problems using technology
• exploration of geometrical propositions using technology

5.0 Standard

At Level 5, students formulate conjectures and follow simple mathematical deductions (for example, if the side length of a cube is doubled, then the surface area increases by a factor of four, and the volume increases by a factor of eight).

Students use variables in general mathematical statements. They substitute numbers for variables (for example, in equations, inequalities, identities and formulas).

Students explain geometric propositions (for example, by varying the location of key points and/or lines in a construction).

Students develop simple mathematical models for real situations (for example, using constant rates of change for linear models). They develop generalisations by abstracting the features from situations and expressing these in words and symbols. They predict using interpolation (working with what is already known) and extrapolation (working beyond what is already known). They analyse the reasonableness of points of view, procedures and results, according to given criteria, and identify limitations and/or constraints in context.

Students use technology such as graphic calculators, spreadsheets, dynamic geometry software and computer algebra systems for a range of mathematical purposes including numerical computation, graphing, investigation of patterns and relations for algebraic expressions, and the production of geometric drawings.

• Mathematical deductions

5.25

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

• development of alternative algebraic models for a set of data and evaluation of their relative merits
• presentation of algebraic arguments using appropriate mathematical symbols and conventions
• evaluation of the appropriateness of the results of their own calculations

5.5

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

• generation of reports from a database by using and, or and not as search tools
• justification or proof of generalisations made from specific cases
• selection and use of technology to explore geometrical and algebraic relationships and data trends

• Verify results from CAS

5.75

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

• use of an ‘equations editor’ to insert mathematical material in a text document
• simulation of events using technology
• representation and manipulation of symbolic expressions using technology
• recognition of functionality of technology and its limitations, such as image resolution, discontinuities in graphs and systematic error in computation through rounding

• Effective and efficient use of a graphics calculator

6.0 Standard

At Level 6, students formulate and test conjectures, generalisations and arguments in natural language and symbolic form (for example, ‘if m² is even then m is even, and if m² is odd then m is odd’). They follow formal mathematical arguments for the truth of propositions (for example, ‘the sum of three consecutive natural numbers is divisible by 3’).

Students choose, use and develop mathematical models and procedures to investigate and solve problems set in a wide range of practical, theoretical and historical contexts (for example, exact and approximate measurement formulas for the volumes of various three dimensional objects such as truncated pyramids). They generalise from one situation to another, and investigate it further by changing the initial constraints or other boundary conditions. They judge the reasonableness of their results based on the context under consideration.

They select and use technology in various combinations to assist in mathematical inquiry, to manipulate and represent data, to analyse functions and carry out symbolic manipulation. They use geometry software or graphics calculators to create geometric objects and transform them, taking into account invariance under transformation.

• Mathematical arguments