Background to the VELS - Mathematics domain

At Level 1, students form small sets of objects from simple descriptions and make simple correspondences between those sets. They count the size of small sets using the numbers 0 to 20. They use one-to-one correspondence to identify when two sets are equal in size and when one set is larger than another. They form collections of sets of equal size. They use ordinal numbers to describe the position of elements in a set from first to tenth. They use materials to model addition and subtraction by the aggregation (grouping together) and disaggregation (moving apart) of objects. They add and subtract by counting forward and backward using the numbers from 0 to 20.

At this level students work on the fundamental principles underlying meaningful counting such as one-to-one correspondence. By counting, comparing and estimating they develop a sense of numbers up to 20 and beyond, they develop facility in counting forwards and backwards by ones and order things within collections.

They use materials and counting strategies with increasing facility to find relationships between small numbers and begin to develop automatic recall of simple number facts including doubles.

MAMA0108 Count, compare and order collections of at least 20 objects.

MAMA0109 Model numbers up to at least 10 and use counting strategies to find and verbalise relationships between small numbers.

MAMA0110 Recognise and write numerals from 1 to 10 and use them to record simple addition and subtraction facts and doubles.

Space
Number
Measurement, chance and data

Structure
Working mathematically

Space
Number
Measurement
Chance and data
Algebra
Reasoning and strategies

At Level 1, students form small sets of objects from simple descriptions and make simple correspondences between those sets. They count the size of small sets using the numbers 0 to 20.

They use one-to-one correspondence to identify when two sets are equal in size and when one set is larger than another.

MAMA0108 Count, compare and order collections of at least 20 objects.

MAMA0109 Model numbers up to at least 10 and use counting strategies to find and verbalise relationships between small numbers.

They form collections of sets of equal size. They use ordinal numbers to describe the position of elements in a set from first to tenth.

They use materials to model addition and subtraction by the aggregation (grouping together) and disaggregation (moving apart) of objects.

They add and subtract by counting forward and backward using the numbers from 0 to 20.

MAMA0110 Recognise and write numerals from 1 to 10 and use them to record simple addition and subtraction facts and doubles.

At Level 1, students recognise, copy and draw points, lines and simple free-hand curves. They identify basic two-dimensional shapes such as triangles, circles and squares and three-dimensional solids and objects such as boxes and balls. They recognise the interior and exterior of shapes and objects. They sort geometric objects according to simple descriptions.

MAMA0101 Recognise and name some simple shapes and objects and use everyday language to describe shape and function.

MAMA0102 Make and draw reasonable representations of simple shapes.

MAMA0103 Copy reasonable representations of simple spatial pictures and patterns.

MAMA0104 Use shape and orientation to fit several simple shapes together by copying or by matching lines.

They place and orientate shapes according to simple descriptions such as next to, beside, in front of, behind, over and under.

They develop and follow simple instructions to move and place shapes and objects in familiar situations in relation to what they can see, and to move themselves from one place to another.

MAMA0107 Represent parts of familiar environments by building models.

MAMA0105 Use and understand simple everyday location words to follow and give an oral direction.

MAMA0106 Follow short paths on simple drawings and models.

At Level 1, students compare length, area, capacity and mass of familiar objects using descriptive terms such as longer, taller, larger, holds more and heavier. They make measurements using informal units such as paces for length, handprints for area, glasses for capacity, and bricks for weight.

MAMA0113 Identify attributes of objects and describe those attributes in the everyday language of measurement.

MAMA0114 Estimate, measure and compare the size of objects using informal methods.

They recognise the continuity of time and the natural cycles such as day/night and the seasons. They correctly sequence days of the week. They use informal units such as heartbeats and hand claps at regular intervals to measure and describe the passage of time.

MAMA0115 Relate time to, and describe time in terms of, familiar recurring phenomena within own life.

MAMA0116 Relate the function of clocks to the telling of time.

They recognise and respond to unpredictability and variability in events, such as getting or not getting a certain number on the roll of a die in a game or the outcome of a coin toss.

MAMA0117 Recognise that some events involve chance. Investigate and describe events that involve chance and recognise the unpredictable nature of particular outcomes.

They collect and display data related to their own activities using simple pictographs.

MAMA0118 Pose questions with guidance and collect information in order to answer the questions posed.

MAMA0119 Represent, summarise and discuss data using concrete and pictorial displays and oral descriptions.

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.

MAMA0120 Recognise what is correct or incorrect and consistent or inconsistent in mathematical situations involving materials, objects, dialogue and, as appropriate, calculators, and correct any inconsistencies encountered.

MAMA0121 Make judgments based on simple criteria.

MAMA0122 Choose an appropriate activity to respond to a mathematical question or represent a situation generated by adults and fellow students.

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.

MAMA0111 Use knowledge of numbers, counting and addition and subtraction relationships to explore and describe simple, everyday numerical situations including the use of money.

MAMA0112 Use materials and a calculator to recognise, generate and represent simple number patterns.

At Level 2, students model the place value of the natural numbers from 0 to 1000. They order numbers and count to 1000 by 1s, 10s and 100s. Students skip count by 2s, 4s and 5s from 0 to 100 starting from any natural number. They form patterns and sets of numbers based on simple criteria such as odd and even numbers. They order money amounts in dollars and cents and carry out simple money calculations. They describe simple fractions such as one half, one third and one quarter in terms of equal sized parts of a whole object, such as a quarter of a pizza, and subsets such as half of a set of 20 coloured pencils.

MANUN202 Model, represent and understand numbers up to 999.

MANUN203 Use informal fraction language in relation to objects and collections of objects.

MANUN201 Count forwards and backwards to and from 1000 and skip-count to 100.

MANUP201 Recognise patterns in the whole number system.

They add and subtract one- and two-digit numbers by counting on and counting back. They mentally compute simple addition and subtraction calculations involving one- or two-digit natural numbers, using number facts such as complement to 10, doubles and near doubles.

MANUM201 Calculate mentally with numbers up to approximately 20.

MANUM202 Use estimation strategies to assist counting and computations when dealing with numbers greater than 20.

They describe and calculate simple multiplication as repeated addition, such as 3 × 5 = 5 + 5 + 5; and division as sharing, such as 8 shared between 4. They use commutative and associative properties of addition and multiplication in mental computation (for example, 3 + 4 = 4 + 3 and 3 + 4 + 5 can be done as 7 + 5 or 3 + 9).

MANUC202 Create and solve number sentences arising from number stories and situations which involve a single operation of addition, subtraction, multiplication or division.

MANUP202 Represent, identify, extend and devise whole number patterns.

At Level 2, students recognise lines, surfaces and planes, corners and boundaries; familiar two-dimensional shapes including rectangles, rhombuses and hexagons, and three-dimensional shapes and objects including pyramids, cones, and cylinders. They arrange a collection of geometric shapes, such as a set of attribute blocks, into subsets according to simple criteria, and recognise when one set of shapes is a subset of another set of shapes.

MASPS201 Identify, name and use common terms to describe features of simple shapes and objects.

MASPS202 Compare and classify shapes and objects using simple spatial criteria.

They recognise and describe symmetry, asymmetry, and congruence in these shapes and objects. They accurately draw simple two-dimensional shapes by hand and construct, copy and combine these shapes using drawing tools and geometry software.

MASPS203 Construct recognisable representations of shapes seen or described.

MASPS204 Make simple spatial pictures, patterns and constructions from verbal and visual instructions.

MASPS206 Use shape, orientation and symmetry to complete simple pictures or patterns.

They apply simple transformations to shapes (flips, turns, slides and enlargements) and depict both the original and transformed shape together.

MASPS205 Describe and explain the effect of simple flips, slides and turns on shapes.

They specify location as a relative position, including left and right, and interpret simple networks, diagrams and maps involving a small number of points, objects or locations.

MASPL201 Use and understand everyday location words to follow and give oral directions.

MASPL202 Use ‘left’ and ‘right’ to describe the position of objects in relation to self.

MASPL203 Locate, follow and orally describe paths on simple maps, models and mazes.

MASPL204 Locate key features when interpreting or making simple maps or models of familiar locations.

At Level 2, students make, describe and compare measurements of length, area, volume, mass and time using informal units.

MAMDM201 Choose and use the appropriate attribute when responding to measurement questions and select units of measure which relate well to the attribute.

MAMDU201 Describe and represent different attributes of objects and distinguish between these in measurement contexts.

They recognise the differences between non-uniform measures, such as hand-spans, to measure length, and uniform measures, such as icy-pole sticks. They judge relative capacity of familiar objects and containers by eye and make informal comparisons of weight by hefting. They describe temperature using qualitative terms (for example, cold, warm, hot). Students use formal units such as hour and minute for time, litre for capacity and the standard units of metres, kilograms and seconds.

MAMDM202 Use everyday language to describe and compare distances, mass, capacity and area.

MAMDM203 Make comparisons of the relative size of two or more objects.

MAMDM204 Use uniform informal units to estimate, measure, compare and order the sizes of objects.

MAMDM205 Measure objects by comparing to formal units and standard units of measurement and using simple, common measuring tools.

Students recognise the key elements of the calendar and place in sequence days, weeks and months. They describe common and familiar time patterns and such as the time, duration and day of regular sport training and tell the time to hours and half-hours using an analog clock, and to hours and minutes using a digital clock.

MAMDT201 Describe, order and sequence events with respect to the time of day or time of the year they usually occur.

MAMDT202 Tell the time using analogue and digital clocks and describe time elapsed (duration) in everyday language and common units of time.

Students predict the outcome of chance events, such as the rolling of a die, using qualitative terms such as certain, likely, unlikely and impossible.

MAMDC201 Classify events as certain, possible, impossible, likely or unlikely.

MAMDC202 Compare two familiar, easily understood events and decide which is more likely to happen.

They collect simple categorical and numerical data (count of frequency) and present this data using pictographs and simple bar graphs.

MAMDD201 Pose questions of interest arising from familiar situations or from collected information.

MAMDD203 Record, represent and summarise data in lists and simple graphs.

MAMDD204 Describe and interpret data in lists and simple graphs.

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.

MAMDD202 Collect and classify data in the form of objects, pictures, statements and numbers to answer questions or test conjectures of interest.

MARSR202 Make and explain judgments based on simple criteria.

MARSS201 Ask and respond to questions which clarify the essential nature of a story, task or problem and identify key information in familiar situations.

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 × key, to check the accuracy of mental and written estimations and approximations and solutions to simple number sentences and equations.

MANUN204 Use a four-function calculator to aid and explore counting and place-value concepts.

MANUC201 Model addition and subtraction operations and use informal written methods based on place-value to solve these problems, checking solutions by estimation and calculator use.

MANUC203 Read, write and interpret symbolic number sentences involving one operation.

MANUP203 Construct and complete simple statements of equality (equations).

MARSR201 Check the truth of number statements and conjectures as well as the consistency of number patterns arising from operations carried out with materials and informal written methods.

MARSS202 Use simple strategies to explore tasks and solve problems.

At Level 3, students use place value (as the idea that ‘ten of these is one of those’) to determine the size and order of whole numbers to tens of thousands, and decimals to hundredths. They round numbers up and down to the nearest unit, ten, hundred, or thousand. They develop fraction notation and compare simple common fractions such as ¾ > 2/3 using physical models.

MANUN301 Recognise the structure of whole numbers up to 5 digits, including place value.

MANUN303 Represent, find, compare and order fractional parts of objects and collections of objects.

They skip count forwards and backwards, from various starting points using multiples of 2, 3, 4, 5, 10 and 100.

They estimate the results of computations and recognise whether these are likely to be over-estimates or under-estimates. They compute with numbers up to 30 using all four operations. They provide automatic recall of multiplication facts up to 10 × 10.

They devise and use written methods for:

• whole number problems of addition and subtraction involving numbers up to 999
• multiplication by single digits (using recall of multiplication tables) and multiples and powers of ten (for example, 5 × 100, 5 × 70)
• division by a single-digit divisor (based on inverse relations in multiplication tables).

MANUN302 Skip-count by numbers of increasing size.

MANUN304 Use decimal notation to represent and compare simple decimal fractions including those resulting from calculator computations.

MANUC301 Use knowledge of place-value to solve and record solutions to addition, subtraction, multiplication and division problems.

MANUM301 Recall or mentally determine basic multiplication and division facts.

MANUM302 Use place-value ideas and the properties of numbers and operations to assist mental computation.

They devise and use algorithms for the addition and subtraction of numbers to two decimal places, including situations involving money. They add and subtract simple common fractions with the assistance of physical models.

MANUC302 Select the appropriate operations and computation methods to solve problems involving whole numbers and money.

MANUC303 State equivalence statements and addition and subtraction facts involving simple common fractions and carry out calculations involving tenths and hundredths.

At Level 3, students recognise and describe the directions of lines as vertical, horizontal or diagonal. They recognise angles are the result of rotation of lines with a common end-point.

MASPS301 Recognise, describe and represent straight, curved, diagonal, horizontal and vertical lines, and angles as rotations of lines.

They recognise and describe polygons. They recognise and name common three-dimensional shapes such as spheres, prisms and pyramids. They identify edges, vertices and faces. They use two-dimensional nets, cross-sections and simple projections to represent simple three-dimensional shapes. They follow instructions to produce simple tessellations (for example, with triangles, rectangles, hexagons) and puzzles such as tangrams.

MASPS302 Use simple conventional spatial language when describing shapes, parts of shapes, objects, parts of objects and simple cross-sections.

MASPS303 Explain and compare the spatial properties of shapes and objects.

MASPS305 Interpret, recognise and name three-dimensional objects from drawings and photographs and make recognisable models and sketches of simple shapes and objects.

They locate and identify places on maps and diagrams. They give travel directions and describe positions using simple compass directions (for example, N for North) and grid references on a street directory.

MASPL301 Use and understand conventional location language to follow and give directions and describe position.

MASPL302 Visualise, find and compare alternative paths on simple maps, grids and mazes.

At Level 3, students estimate and measure length, area, volume, capacity, mass and time using appropriate instruments.

MAMEM301 Make increasingly accurate estimates of measurements using informal units and standard units.

They recognise and use different units of measurement including informal (for example, paces), formal (for example, centimetres) and standard metric measures (for example, metre) in appropriate contexts. They read linear scales (for example, tape measures) and circular scales (for example, bathroom scales) in measurement contexts. They read digital time displays and analogue clock times at five-minute intervals. They interpret timetables and calendars in relation to familiar events.

MAMEM302 Measure and compare using appropriate informal units.

MAMEM303 Estimate and accurately measure length, mass, volume and temperature using formal units and standard units.

MAMET301 Estimate short and long periods of time, describe duration of time, and make and use timetables, schedules and calendars.

MAMET302 Tell the time using digital and analogue clocks.

They compare the likelihood of everyday events (for example, the chances of rain and snow). They describe the fairness of events in qualitative terms. They plan and conduct chance experiments (for example, using colours on a spinner) and display the results of these experiments.

MACDC301 Identify and record outcomes from simple chance experiments.

MACDC302 Compare and order the likelihood of outcomes of simple chance experiments and of everyday events, and choose appropriate methods for random selection.

MACDP302 Determine appropriate procedures to collect information relevant to questions and conjectures of interest.

They recognise different types of data: non-numerical (categories), separate numbers (discrete), or points on an unbroken number line (continuous).They use a column or bar graph to display the results of an experiment (for example, the frequencies of possible categories).

MACDS301 Organise and summarise category and whole number data.

MACDS303 Use graphical methods involving scale to display discrete and continuous data.

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.

MANUC301 Use knowledge of place-value to solve and record solutions to addition, subtraction, multiplication and division problems.

MANUP301 Use rules involving addition, subtraction and multiplication to devise, describe, extend and test number patterns.

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).

MANUC303 State equivalence statements and addition and subtraction facts involving simple common fractions and carry out calculations involving tenths and hundredths.

MANUP302 Detect similarities and differences in the nature of the operations of addition, subtraction and multiplication.

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.

MACDS302 Use diagrams and two-way tables to summarise and display discrete data.

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.

MANUP303 Construct and complete simple statements of equality involving whole numbers and fractions.

At Level 3, students apply number skills to everyday contexts such as shopping, with appropriate rounding to the nearest five cents.

MARSR301 Make and test simple conjectures.

MANUC302 Select the appropriate operations and computation methods to solve problems involving whole numbers and money.

MARSR302 Make judgments about the accuracy of reasoning and results.

MARSS301 Generate mathematical questions from presented data and from familiar contexts.

They recognise the mathematical structure of problems and use appropriate strategies (for example, recognition of sameness, difference and repetition) to find solutions.

MARSS302 Clarify the essential nature of a task or problem and identify key information in familiar situations.

MARSS303 Use familiar representations, processes and concepts to explore unfamiliar tasks and problems.

MASPS307 Identify symmetry in regular two-dimensional shapes.

MASPS304 Visualise and describe some of ‘what is not seen’ of simple objects.

MASPL303 Interpret and describe location and direction using grid references and cardinal compass points.

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).

MARSS304 Use the guess–check–improve process in appropriate contexts.

MAMEU301 Describe the relationship between attributes.

MACDP301 Identify information required to answer questions or test conjectures, refining the questions where necessary.

MACDP303 Modify the method of data collection and classification to refine a question or investigate a further question. 

Students use calculators to explore number patterns and check the accuracy of estimations.

MANUM303 Make estimates to check the reasonableness of the results of written computation and calculator use.

They use a variety of computer software to create diagrams, shapes, tessellations and to organise and present data.

MASPS306 Copy and create simple patterns involving translating, rotating and reflecting multiple copies of a shape and informally describe the transformations used.

MASPS307 Identify symmetry in regular two-dimensional shapes.

At Level 4, students comprehend the size and order of small numbers (to thousandths) and large numbers (to millions). They model integers (positive and negative whole numbers and zero), common fractions and decimals. They place integers, decimals and common fractions on a number line. They create sets of number multiples to find the lowest common multiple of the numbers. They interpret numbers and their factors in terms of the area and dimensions of rectangular arrays (for example, the factors of 12 can be found by making rectangles of dimensions 1 × 12, 2 × 6, and 3 × 4).

MANUN401 Use place-value knowledge to read, write and order negative whole numbers and decimal numbers from thousandths to millions.

MANUN402 Compare and order common fractions.

MANUN503 Compare and order negative numbers.

MANUP402 Specify multiples and factors of whole numbers.

Students identify square, prime and composite numbers. They create factor sets (for example, using factor trees) and identify the highest common factor of two or more numbers. They recognise and calculate simple powers of whole numbers (for example, 24 = 16).

Students use decimals, ratios and percentages to find equivalent representations of common fractions (for example, 3/4 = 9/12 = 0.75 = 75% = 3 : 4 = 6 : 8). They explain and use mental and written algorithms for the addition, subtraction, multiplication and division of natural numbers (positive whole numbers).

They add, subtract, and multiply fractions and decimals (to two decimal places) and apply these operations in practical contexts, including the use of money.

MANUN403 Rename common fractions as decimals and percentages.

MANUM401 Recall automatically basic multiplication and division facts, simple common fraction facts and frequently used common fraction, decimal and percentage equivalences.

MANUM402 Use knowledge of place-value and number properties to increase the range of computations which can be carried out mentally.

MANUC401 Use written methods to add and subtract decimal numbers.

MANUC402 Use written methods to multiply and divide whole numbers.

MANUC403 Use models to illustrate the four operations with common fractions, and develop written methods for carrying out these operations.

They use estimates for computations and apply criteria to determine if estimates are reasonable or not.

MANUM403 Use estimation strategies to check the results of written or calculator computations.

At Level 4, students classify and sort shapes and solids (for example, prisms, pyramids, cylinders and cones) using the properties of lines (orientation and size), angles (less than, equal to, or greater than 90°), and surfaces. They create two-dimensional representations of three dimensional shapes and objects found in the surrounding environment.

MASPS401 Recognise, describe and represent parallel, perpendicular, horizontal and vertical lines, right angles, and angles greater than or less than 90 degrees (multiples of 45 degrees).

MASPS404 Draw conventional representations of prisms, pyramids, cylinders and cones.

They develop and follow instructions to draw shapes and nets of solids using simple scale. They describe the features of shapes and solids that remain the same (for example, angles) or change (for example, surface area) when a shape is enlarged or reduced. They apply a range of transformations to shapes and create tessellations using tools (for example, computer software).

MASPL404 Interpret formal maps and make detailed maps and plans.

MASPL405 Use a simple scale (for example, 1 centimetre for each metre) when making, interpreting and using maps and plans.

MASPS407 Enlarge (or reduce) two-dimensional shapes and simple three-dimensional objects.

Students use the ideas of size, scale, and direction to describe relative location and objects in maps. They use compass directions, coordinates, scale and distance, and conventional symbols to describe routes between places shown on maps. Students use network diagrams to show relationships and connectedness such as a family tree and the shortest path between towns on a map.

MASPS405 Visualise, explain and represent ‘what is not seen’ of an object.

MASPL401 Use and understand conventional location language including distance and direction.

MASPL402 Use informal coordinate systems (positive numbers only) and intermediate compass points to specify location or give directions.

MASPL403 Visualise and find paths to satisfy specifications on maps, grids and mazes.

MAMEM403 Draw and construct objects using accurate measurements.

At Level 4, students use metric units to estimate and measure length, perimeter, area, surface area, mass, volume, capacity time and temperature. They measure angles in degrees. They measure as accurately as needed for the purpose of the activity. They convert between metric units of length, capacity and time (for example, L–mL, sec–min).

MAMEM401 Choose attributes and standard units appropriate to the task.

MAMEM402 Make judgments about the relative size of objects based on comparison to known benchmarks or standard units.

MAMEM404 Use measuring instruments, reading simple scales and measuring accurately to the nearest marked gradation, taking into account the degree of exactness required.

MAMET401 Use and construct timetables and use and analyse calendars.

MAMET402 Estimate, measure and calculate time elapsed (duration).

MAMET403 Tell the time accurately using analogue clocks and digital clocks.

Students describe and calculate probabilities using words, and fractions and decimals between 0 and 1. They calculate probabilities for chance outcomes (for example, using spinners) and use the symmetry properties of equally likely outcomes. They simulate chance events (for example, the chance that a family has three girls in a row) and understand that experimental estimates of probabilities converge to the theoretical probability in the long run.

MACDC401 Examine the outcomes from simple chance experiments and data on familiar events to order outcomes and events from least to most likely.

MACDC402 Use and interpret numerical statements which quantify chance. MACDC403 Use language of chance in everyday situations.

Students recognise and give consideration to different data types in forming questionnaires and sampling. They distinguish between categorical and numerical data and classify numerical data as discrete (from counting) or continuous (from measurement).

MACDS401 Prepare tabular displays of discrete and continuous data.

MACDP402 Collect and record data systematically.

MACDS403 Compare, order and summarise data sets using simple numerical methods.

MACDI401 Extract and interpret numerical information contained in tables, data displays and databases.

MACDP501 Decide the nature of data required to effectively answer specific questions and plan ways to collect and organise it.

They present data in appropriate displays (for example, a pie chart for eye colour data and a histogram for grouped data of student heights). They calculate and interpret measures of centrality (mean, median, and mode) and data spread (range).

MACDS402 Prepare visual displays of discrete and continuous (measurement) data using a range of graphical methods.

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’).

MASPS402 Analyse, explain and compare the spatial properties of lines, angles, polygons, polyhedra and cross-sections using conventional spatial terms.

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)).

MANUP401 Generate and investigate number sequences which may involve fractions, decimals and combinations of operations, using a calculator where appropriate.

MAMEU402 Investigate the relationship between area and perimeter and calculate the area of a polygon.

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.

MANUP403 Construct, verify and complete number sentences involving the four operations, brackets, decimal numbers and fractions.

MASPS406 Visualise, test and describe transformations of shapes.

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).

MACDP401 Design and prepare surveys and experiments to answer questions or test conjectures and predictions.

MARSR401 Make and test simple conjectures in each mathematics strand.

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).

MACDI402 Interpret, discuss and compare data displays, including how well they communicate information.

MARSR403 Use and interpret simple mathematical models.

MARSS401 Generate mathematical questions from presented data and from familiar contexts.

MARSS402 Clarify the essential nature of a task or problem and identify key information in the context under consideration.

MARSS403 Use a range of strategies for inquiry when responding to tasks and problems.

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.

MANUC404 Analyse a problem situation which may involve several different operations, decimal numbers, negative whole numbers and common fractions; express the problem symbolically and choose appropriate computational methods to solve it.

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.

MANUM403 Use estimation strategies to check the results of written or calculator computations. MASPS403 Make congruent copies of given three-dimensional objects. MARSR402 Make judgments about the accuracy of reasoning and results and modify working accordingly.

At Level 5, students identify complete factor sets for natural numbers and express these natural numbers as products of powers of primes (for example, 36 000 = 2⁵ × 3² × 5³).

Students express natural numbers base 10 in binary form, (for example, 42₁₀ = 101010₂), and add and multiply natural numbers in binary form (for example, 101₂ + 11₂ = 1000₂ and 101₂ × 11₂ = 1111₂).

Students understand ratio as both set: set comparison (for example, number of boys : number of girls) and subset: set comparison (for example, number of girls : number of students), and find integer proportions of these, including percentages (for example, the ratio number of girls: the number of boys is 2 : 3 = 4 : 6 = 40% : 60%).

MANUN501 Compare and order common and decimal fractions, percentages and ratios.

MANUN502 Find prime factors and understand the use of whole-number powers and the square root sign.

They write equivalent fractions for a fraction given in simplest form (for example, 2/3 = 4/6 = 6/9 = … ). They know the decimal equivalents for the unit fractions 1/2, 1/3, 1/4, 1/5, 1/8, 1/9 and find equivalent representations of fractions as decimals, ratios and percentages (for example, a subset: set ratio of 4:9 can be expressed equivalently as 4/9 = 0.4 ≈ 44.44%). They write the reciprocal of any fraction and calculate the decimal equivalent to a given degree of accuracy.

MANUN501 Compare and order common and decimal fractions, percentages and ratios.

Students use knowledge of perfect squares when calculating and estimating squares and square roots of numbers (for example, 20² = 400 and 30² = 900 so √700 is between 20 and 30). They evaluate natural numbers and simple fractions given in base-exponent form (for example, 5⁴ = 625 and (2/3)² = 4/9). They know simple powers of 2, 3, and 5 (for example, 2⁶ = 64, 3⁴ = 81, 5³ = 125). They calculate squares and square roots of rational numbers that are perfect squares (for example, √0.81 = 0.9 and √9/16 = 3/4). They calculate cubes and cube roots of perfect cubes (for example, ³√64 = 4). Using technology they find square and cube roots of rational numbers to a specified degree of accuracy (for example, ³√200 = 5.848 to three decimal places).

MANUM501 Extend the use of basic number facts to mentally compute operations on fractions and decimals, and squares and square roots.

MANUC501 Use written methods to carry out the four operations on decimal numbers.

MANUC502 Use written methods to carry out the four operations on common fractions and decimals.

MANUC503 Carry out the four operations in cases where both positive and negative integers are involved.

Students use a range of strategies for approximating the results of computations, such as front-end estimation and rounding
(for example, 925 ÷ 34 ≈ 900 ÷ 30 = 30).

Students use efficient mental and/or written methods for arithmetic computation involving rational numbers, including division of integers by two-digit divisors. They use approximations to π in related measurement calculations
(for example, π × 5² = 25π = 78.54 correct to two decimal places).

They use technology for arithmetic computations involving several operations on rational numbers of any size.

MANUM503 Use estimation strategies to check computations with fractions and decimals.

At Level 5, students construct two-dimensional and simple three-dimensional shapes according to specifications of length, angle and adjacency. They use the properties of parallel lines and transversals of these lines to calculate angles that are supplementary, corresponding, allied (co-interior) and alternate.

MASPS501 Use appropriate geometric techniques such as paper-folding, ruler/compasses/protractor, and a computer drawing package to accurately draw simple two-dimensional shapes, attending to essential details (for example, matching lengths and angles).

MASPS502 Make and analyse models of solid objects.

MASPS503 Establish congruence by superimposition, including cases involving rotation and reflection.

MASPS504 Make designs that exhibit symmetries.

MASPS505 Enlarge (or reduce) two-dimensional figures or three-dimensional objects using scale factors.

They describe and apply the angle properties of regular and irregular polygons, in particular, triangles and quadrilaterals. They use two-dimensional nets to construct a simple three-dimensional object such as a prism or a platonic solid. They recognise congruence of shapes and solids. They relate similarity to enlargement from a common fixed point. They use single-point perspective to make a two-dimensional representation of a simple three-dimensional object. They make tessellations from simple shapes.

MASPS506 Use angle relations involving transversals and pairs of parallel lines to solve problems, giving a reason for the solution.

MASPS507 Investigate and apply properties of regular and irregular polygons and circles.

Students use coordinates to identify position in the plane. They use lines, grids, contours, isobars, scales and bearings to specify location and direction on plans and maps. They use network diagrams to specify relationships. They consider the connectedness of a network, such as the ability to travel through a set of roads between towns.

MASPL501 Draw and interpret diagrams representing familiar situations.

MASPL502 Use coordinates in 4 quadrants, grids and bearings to specify location of points.

MASPL503 Use information on a map to specify and obtain distances, heights and directions.

At Level 5, students measure length, perimeter, area, surface area, mass, volume, capacity, angle, time and temperature using suitable units for these measurements in context. They interpret and use measurement formulas for the area and perimeter of circles, triangles and parallelograms and simple composite shapes. They calculate the surface area and volume of prisms and cylinders.

MAMEM501 Recognise and select appropriate metric units and levels of accuracy for measuring quantities and rates.

MAMEM502 Select, use and adapt instruments to measure length, mass, capacity, volume, angle and temperature.

MAMET501 Measure, estimate and calculate time and duration of time.

MAMEU501 Obtain areas by counting squares in order to develop new rules for the area of regular shapes.

MAMEU502 Develop and use rules to calculate perimeters of polygons and circles, areas of shapes based on triangles, rectangles and circles, and volumes and surface areas of rectangular prisms.

MAMEU503 Calculate and use rates.

Students estimate the accuracy of measurements and give suitable lower and upper bounds for measurement values. They calculate absolute percentage error of estimated values.

MAMEM503 Use judgments of the size of metric units to make and refine estimates of quantities.

Students use appropriate technology to generate random numbers in the conduct of simple simulations.

Students identify empirical probability as long-run relative frequency. They calculate theoretical probabilities by dividing the number of possible successful outcomes by the total number of possible outcomes. They use tree diagrams to investigate the probability of outcomes in simple multiple event trials.

MACDC501 Analyse experiments to determine the theoretical probability of events.

MACDC502 Carry out experiments involving chance to estimate the probability of events and to simulate situations.

Students organise, tabulate and display discrete and continuous data (grouped and ungrouped) using technology for larger data sets. They represent uni-variate data in appropriate graphical forms including dot plots, stem and leaf plots, column graphs, bar charts and histograms. They calculate summary statistics for measures of centre (mean, median, mode) and spread (range, and mean absolute difference), and make simple inferences based on this data.

MACDS501 Present collected data in tables, databases and spreadsheets. MACDS502 Present univariate data using graphical techniques and technology.
MACDS503 Summarise a data set by obtaining measures of central location and spread ‘by hand’ and by using technology. MACDI501 Interpret and evaluate information contained in tables, visual displays and databases and report on methods of data collection. MACDI502 Interpret simple measures of location and spread and use them in comparisons. MACDI503 Draw inferences from collected or published data. MACDI504 Make predictions on the basis of samples. MACDP502 Obtain data related to an area of interest.

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.

MAALE503 Construct and interpret rules for simple relationships between variables and between successive terms in sequences.

MARSR502 Make judgments about the quality of the reasoning in a mathematical argument expressed verbally or in symbolic form.

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 = 2⁸, 2⁰ = 1, 2³ ÷ 2⁵ = 2⁻², (5²)³ = 5⁶ and (3 × 4)² = 3² × 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, 1/2 + −1/2 = 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 = πr² then r = √A/πfor r > 0).

MANUM502 Use properties of numbers to carry out mental computations involving whole numbers, decimals and common fractions.

MAALE501 Develop, interpret and simplify mathematical expressions which describe rules for relationships and mensuration formulas.

They solve simple equations (for example, 5x + 7 = 23, 1.4x − 1.6 = 8.3, and 4x² − 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 x² > 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.

MAALI501 Develop linear and other simple equations and inequalities from information provided in a given context.

MAALI502 Make judgments about, and verify truth values for propositions expressed as linear equalities and inequalities.

MAALF502 Sketch and interpret graphs of linear and other simple relationships.

MAALF503 Plot graphs of linear and other simple functions and use linear functions to model data.
MAALF501 Use ordered pairs to locate and describe the positions of points on a Cartesian coordinate grid.

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).

MARSR503 Use and interpret simple mathematical models and make judgments about the accuracy and suitability of the results obtained by using mathematical models.

MARSS501 Generate mathematical questions for inquiry from presented data, familiar contexts and the experience gained from inquiries in relation to previous tasks and problems.

MARSR501 Make, test and modify conjectures.

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.

MARSS502 Clarify the essential nature of a task or problem and identify key information in familiar and unfamiliar situations.

MARSS503 Apply a range of strategies for inquiry to complete tasks and solve problems.

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.

MANUC504 Select and use an appropriate sequence of operations and appropriate computation methods to solve problems.

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.

MAALE502 Demonstrate the equivalence (identity) or difference between simple algebraic expressions.

At Level 6, students comprehend the set of real numbers containing natural, integer, rational and irrational numbers. They represent rational numbers in both fractional and decimal (terminating and infinite recurring) forms
(for example, 14/25 = 1.16, 0.47 = 47/99). They comprehend that irrational numbers have an infinite non-terminating decimal form. They specify decimal rational approximations for square roots of primes, rational numbers that are not perfect squares, the golden ratio φ , and simple fractions of π correct to a required decimal place accuracy.
Students use the Euclidean division algorithm to find the greatest common divisor (highest common factor) of two natural numbers (for example, the greatest common divisor of 1071 and 1029 is 21 since 1071 = 1029 × 1 + 42, 1029 = 42 × 24 + 21 and 42 = 21 × 2 + 0).

MANUN601 Move freely between the fraction, percentage and decimal forms of rational numbers.

MANUN602 Represent irrational square roots appropriately.

MANUN603 Interpret and use numbers written with integer powers.

MANUN604 Work with the real number system as the union of rational numbers and irrational numbers.

Students carry out arithmetic computations involving natural numbers, integers and finite decimals using mental and/or written algorithms (one- or two-digit divisors in the case of division). They perform computations involving very large or very small numbers in scientific notation (for example, 0.0045 × 0.000028 = 4.5 × 10⁻³ × 2.8 × 10⁻⁵ = 1.26 × 10⁻⁷).

MANUM601 Use automatic recall of decimal and percentage equivalents of common fractions and properties of numbers to mentally compute percentages of quantities.

MANUC601 Use appropriate methods of computation to carry out the four operations on, and evaluate powers and roots of, common and decimal fractions and numbers expressed in index and surd notation.

MANUC602 Solve problems involving rates, ratios and percentages.

MANUC603 Solve problems involving direct or partial variation.

They carry out exact arithmetic computations involving fractions and irrational numbers such as square roots
(for example, √18 = 3√2, √(3/2) = (√6)/2) and multiples and fractions of π (for example π + π/4 = 5/4). They use appropriate estimates to evaluate the reasonableness of the results of calculations involving rational and irrational numbers, and the decimal approximations for them. They carry out computations to a required accuracy in terms of decimal places and/or significant figures.

MANUM602 Use place-value and index notation to mentally compute powers of multiples of 10, roots and multiples of 100 and check the results of calculator computations.

MANUN605 Interpret and represent numbers using fractional powers and surd notation.

At Level 6, students represent two- and three-dimensional shapes using lines, curves, polygons and circles. They make representations using perspective, isometric drawings, nets and computer-generated images. They recognise and describe boundaries, surfaces and interiors of common plane and three-dimensional shapes, including cylinders, spheres, cones, prisms and polyhedra.

MASPS601 Use appropriate geometric tools and knowledge of properties to construct parallels, perpendiculars, angles, and simple polygons.

MASPS602 Describe and compare common representations of three-dimensional shapes.

MASPS603 Construct and analyse properties of selected polyhedra.

MASPS608 Use knowledge of geometric properties to construct accurate representations in two and three dimensions.

Students explore the properties of spheres.

Students use the conditions for shapes to be congruent or similar. They apply isometric and similarity transformations of geometric shapes in the plane. They identify points that are invariant under a given transformation (for example, the point (2, 0) is invariant under reflection in the x-axis, so the x axis intercept of the graph of y = 2x − 4 is also invariant under this transformation). They determine the effect of changing the scale of one characteristic of two- and three-dimensional shapes (for example, side length, area, volume and angle measure) on related characteristics.
They recognise the features of circles (centre, radius, diameter, chord, arc, semi-circle, circumference, segment, sector and tangent) and use associated angle properties.

MASPS604 Investigate what properties of two-dimensional shapes are changed by enlargement/reduction, or what properties are not changed.

MASPS605 Use congruence and similarity conditions for triangles to solve mathematical and practical problems.

MASPS607 Identify and use geometric terms and features relating to circles, such as diameter, radius, chord, tangent, arc, sector, segment.

MASPS609 Choose and apply appropriate scales for making scale drawings of two and three dimensional shapes.

MAMEU603 Use similarity relationships and scale factors in and between figures.

MAMEU606 Investigate similarity relationships in and between shapes and solids.

They use latitude and longitude to locate places on the Earth’s surface and measure distances between places using great circles.

Students describe and use the connections between objects/location/events according to defined relationships (networks).

MASPL601 Visualise, sketch and describe paths of points that move subject to definite constraints.

MASPL602 Investigate and describe the locus, that is, the path of an object moving in two-dimensional and three-dimensional space.

MASPL603 Specify directions using lengths, compass directions and bearings.

MASPL604 Construct and investigate the locus (path) of an object moving according to a rule expressed in conventional mathematical language.

MASPL605 Interpret and use coordinates and graphs in describing a locus (path).

MASPL606 Investigate pathways which satisfy known constraints.

At Level 6, students estimate and measure length, area, surface area, mass, volume, capacity and angle. They select and use appropriate units, converting between units as required. They calculate constant rates such as the density of substances (that is, mass in relation to volume), concentration of fluids, average speed and pollution levels in the atmosphere. Students decide on acceptable or tolerable levels of error in a given situation. They interpret and use mensuration formulas for calculating the perimeter, surface area and volume of familiar two- and three-dimensional shapes and simple composites of these shapes. Students use pythagoras’ theorem and trigonometric ratios (sine, cosine and tangent) to obtain lengths of sides, angles and the area of right-angled triangles.

They use degrees and radians as units of measurement for angles and convert between units of measurement as appropriate.

MAMEM601 Choose units, measurements and levels of accuracy appropriate to a measurement situation.

MAMET601 Calculate time intervals and plan daily schedules to meet special requirements.

MAMEU601 Calculate using rates and interpret graphs involving rates.

MAMEM603 Measure, estimate and develop approximations for areas and volumes of irregular and complex shapes.

MAMET602 Use appropriate units for short and long time intervals.

MAMEM602 Use estimates where appropriate and judge the reasonableness of these estimates.

MAMEU602 Use length, area and volume relationships involving triangles, quadrilaterals, circles, prisms and pyramids.

MAMEU604 Use trigonometry and Pythagoras’ theorem.

MAMEM604 Choose and use standard mks units, including derived units.

MAMEM605 Identify imprecision in measurements and estimate uncertainty in measures judging their acceptability for use.

MAMEU605 Use mensuration formulas directly and indirectly for lengths, angles, areas, surface areas and volumes of shapes and solids.

Students estimate probabilities based on data (experiments, surveys, samples, simulations) and assign and justify subjective probabilities in familiar situations. They list event spaces (for combinations of up to three events) by lists, grids, tree diagrams, venn diagrams and karnaugh maps (two-way tables).

Students comprehend the difference between a population and a sample. They generate data using surveys, experiments and sampling procedures. They calculate summary statistics for centrality (mode, median and mean), spread (box plot, inter-quartile range, outliers) and association (by-eye estimation of the line of best fit from a scatter plot). They distinguish informally between association and causal relationship in bi-variate data, and make predictions based on an estimated line of best fit for scatter-plot data with strong association between two variables.

MACDP601 Design, trial and refine experiments, surveys and simulations.

MACDP602 Collect and record bivariate and time series data.

MACDP603 Identify variables and use these to set up fields in simple databases.

MACDS601 Organise and group raw data, using equal interval widths for continuous data sets.

MACDS602 Construct graphical displays for grouped univariate data and compare two sets of data.

MACDS603 Calculate measures of central location and spread and identify outliers.

MACDS604 Construct graphical displays for bivariate data and time series data.

MACDI604 Informally interpret relationships in bivariate data.

MACDP604 Plan and conduct experiments, simulations and surveys.

MACDP605 Use a variety of sampling methods with some consideration of sample size.

MACDS606 Display and summarise univariate data to show location and variability and to provide comparisons when relevant.

MACDS607 Display and summarise bivariate data to indicate association.

MACDI607 Use graphical displays for interpretation of association in bivariate data sets and trends in time series data sets.

They calculate probabilities for complementary, mutually exclusive, and compound events (defined using and, or and not). They classify events as dependent or independent. 

MACDC601 Estimate probabilities and proportions using published data, experiments and simulations.

MACDC602 Analyse situations involving mutually exclusive, complementary and compound events and assign probabilities to those events.

MACDC603 Assign and use odds and subjective probability.

MACDC604 Compare results of simulations using a variety of random number generators with known probability models.

MACDC605 Identify dependent and independent events, and calculate and interpret conditional probabilities.

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, ∩, 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, π, φ 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, 4x − 8 = 2(2x − 4) = 4(x − 2); (2a − 3)² = 4a² − 12a + 9; (3w)³ = 27w³; (x³y /xy² = x²y⁻¹; 4/xy = 2/x × 2/y).

MAALE601 Use methods of algebraic manipulation to rearrange and simplify mathematical expressions and change the subject of a formula.

MAALE602 Use algebraic manipulation to demonstrate the equivalence (identity) or difference between various mathematical expressions over a specified domain.

MAALE605 Identify and use a range of methods of algebraic manipulation to rearrange and simplify mathematical expressions involving several variables or constants.

MAALE606 Use algebraic manipulation to demonstrate the equivalence (identity) or difference between more complex mathematical expressions, such as those involving reciprocal or square root term.

MACDS605 Organise univariate and bivariate data systematically.

MACDI606 Interpret detailed information contained in tables and graphs.

MAALE603 Construct and interpret rules for linear, quadratic, reciprocal and exponential relationships.

MAALE607 Construct and interpret rules for relationships involving both familiar and more complex algebraic expressions.

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) = ax + c, with reference to gradient and y axis intercept, f(x) = a(x + b)² + c and f(x) = ca^x.

MAALF601 Plot and sketch graphs of linear, quadratic and exponential functions and other simple functions.

MAALF603 Develop variation relationships between data involving two variables.

MAALF604 Use linear, quadratic and exponential functions to model data in a variety of contexts.

MAALF605 Draw graphs of familiar and other functions, such as square root, reciprocal, logarithmic and circular functions, by plotting or by sketching using key features of the function involved.

MAALF606 Interpret graphs of familiar functions and other functions such as square root, reciprocal, logarithmic and circular functions.

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.

MAALI601 Develop linear and quadratic equations and inequalities from information provided in a given context.

MAALI602 Use algebraic and graphical techniques to find, verify and interpret in context the solutions for linear and quadratic equations and inequalities.

MAALI603 Find rational approximations to solutions of quadratic and other equations, including the use of iteration to improve on an initial estimate.

MAALF602 Interpret graphs of linear, quadratic, exponential functions and other simple functions.

MAALI605 Use techniques of algebraic manipulation and estimation (including iterations to improve estimates) together with Cartesian graphs to find and verify solutions for equations, inequalities, and simultaneous pairs of equations and inequalities involving linear, quadratic, reciprocal and exponential expressions.

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’).

MAALE604 Use mathematical expressions to construct a convincing argument to justify conjectures and assertions.

MARSR601 Formulate and test conjectures and generalisations.

MARSR602 Interpret and make judgments about the quality of the reasoning in a mathematical argument expressed verbally or in symbolic form.

MAALE608 Select and use linear, quadratic, reciprocal, exponential and other algebraic expressions to prove mathematical propositions.

MARSR604 Formulate and test generalisations.

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).

MANUC606 Use knowledge of direct and inverse variation to solve problems.

MACDI601 Interpret and evaluate information collected from published data or extracted from prepared databases.

MACDI602 Evaluate procedures used in data collection.

MACDI603 Make comparisons between two sets of data.

MARSR603 Make judgments about the accuracy and suitability of the results obtained by using a mathematical model.

MANUM603 Decrease own dependence on the use of calculators and increase ability to check the results of calculations by automatic recall of a range of number facts and approximations.

MACDC606 Identify and explain sources and effects of chance variation.

MAALI604 Formulate equations and inequalities with any associated restrictions on the variables used, from ‘word problems’ and representations in words or graphs of actual and invented situations, and interpret corresponding solutions, or solution sets, with respect to the context of formulation.

MARSR605 Make judgments about the accuracy and suitability of the results obtained by using a range of mathematical models.

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.

MARSS601 Choose and use a range of strategies for inquiry when responding to tasks and problems.

MACDI605 Critically examine the uses made of assembled information and suggest appropriate further action.

MAALF607 Develop variation relationships between data involving two variables, including inverse (reciprocal) variation.

MAALF608 Use familiar functions and other functions such as square root, reciprocal, logarithmic and circular functions to model data in a variety of contexts.

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.

MASP608 Use knowledge of geometrical properties to develop a logical series of steps to justify a geometric theorem. MARSS603 Choose and use a range of strategies for inquiry when responding to tasks and problems generated by self or others.

They use geometry software or graphics calculators to create geometric objects and transform them, taking into account invariance under transformation.

MANUC604 Choose and use appropriate technology to assist with the computations necessary to solve numerical problems at this level.

MANUC605 Choose appropriate levels of accuracy and computational methods to carry out calculations required to solve problems involving real numbers.