Mathematics

	Level 1	Level 2	Level 3	Level 4	Level 5	Level 6
Standards
Learning focus

Learning focus

As students work towards the achievement of Level 1 standards in Mathematics, they manipulate and play with objects to develop links between their immediate environment, everyday language and mathematical activity.

In Number, students manipulate and group physical objects and drawings to develop basic understanding of the concepts of number and numerals. They group objects into sets(collections) and form simple correspondences (relations) between two sets; for example, in sharing pencils among students. They learn to count the number of objects up to 20 and relate the number counted to the use of a numeral. They describe and place objects in order such as first, second and third. They model addition by putting groups of objects together and counting the combined set and they model subtraction by moving apart groups of objects.

In Space, students manipulate and investigate the properties of basic two- and three-dimensional shapes. They use everyday objects and drawings to identify and describe points, lines, edges and surfaces. They recognise inside and outside. They participate in activities in which they create and follow simple verbal instructions to locate items in the classroom and immediate environment.

In Measurement, chance and data, students learn to compare common objects using terms such as longer, heavier, fuller and hotter. They begin to make estimates and simple measurements using informal units such as a number of paper clips in a length. In playing games of chance, students begin to recognise the unpredictability and uncertainty of events such as the roll of a die. They investigate situations requiring data collection and presentation in simple displays such as a pictogram of family pets.

When Working mathematically, students undertake activities and play to develop skills in making correspondences (for example, games such as Memory and activities such as matching students with their birth months). They create and explore number patterns using counters or other objects. They take risks by making and exploring conjectures relating to numbers, patterns, shapes and measurements (for example, ‘the bigger the object the heavier it is’ or ‘the next shape in a sequence will be …’). Students work with calculators to check the results of simple addition and subtraction. They draw and copy simple shapes and patterns by hand and also by using a computer drawing package.

Standards

Number

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.

Space

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. 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.

Measurement, chance and data

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.

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.

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

Working mathematically

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.

Structure

In Mathematics, standards for the Structure dimension are introduced at Level 3.

Learning focus

As students work towards the achievement of Level 2 standards in Mathematics, they begin to use mathematical symbols and language to describe their mathematical explorations of daily life.

In Number, students learn to use base 10 models (units, longs, flats and cubes) and arrays to identify, order and model the counting numbers up to 1000. They create number patterns mentally, by hand and with the use of the constant addition facility of calculators. They use models and arrays to support the development of skip counting up to 100. They recognise patterns created by skip counting (for example, when counting by fours, the pattern of the ones digits is 4, 8, 2, 6, 0, 4, 8). Students perform simple addition (count on) and subtraction (count back) using numbers up to 100. They use equal groups of objects and rectangular arrays to model multiplication and equal sharing for division. Students divide geometric objects including lines, arrays and regular shapes into equal parts to develop the concept of a simple fraction as part of a whole. They learn to order money amounts in dollars and cents, form different totals using dollars and cents, and carry out simple calculations such as change from small amounts.

In Space, students participate in activities which focus on identification of key features of shapes and solids. They learn to name familiar two- and three-dimensional shapes. They draw simple two-dimensional shapes, and visualise and describe the effect of transformations (for example, slides, flips and turns). They use mirrors and folding to investigate symmetry of shapes. Students learn to construct and follow directions, informal maps, diagrams and routes to locations in the local environment.

In Measurement, chance and data, students learn to use both non-uniform (for example, hand-spans) and uniform (for example, pencil length) informal measurement units. They recognise time units (second, minute, hour, day, week, and month) and investigate basic time patterns and cycles. They learn to tell the time using analogue and digital clocks.

Students pose and respond to questions leading to data collection. They use pictographs and bar graphs to organise and present data. They play games of chance to recognise and quantitatively describe the variability of outcomes. They use terms such as unlikely and almost certain, more likely and less likely to describe everyday chance events.

When Working mathematically, students learn to use a combination of everyday language and mathematical statements and symbols to describe their manipulation and play with sets of numbers, shapes, objects and patterns. They model and describe daily activities and familiar events using physical materials, diagrams and maps (for example, use a 1–1 graph to show attendance at class).

Students test the truth of conjectures by attempting to find examples or counter-examples, and exploring special cases.

They develop and consolidate their understanding of the commutative and associative properties for addition and multiplication. They learn to use a calculator to check estimations, computations and solutions to simple number sentences and equations.

Standards

Number

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. 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. 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).

Space

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

Measurement, chance and data

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

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 at hours and half-hours using an analogue clock, and to hours and minutes using a digital clock.

Students predict the outcome of chance events, such as the rolling of a die, using qualitative terms such as certain, likely, unlikely and impossible. They collect simple categorical and numerical data (count of frequency) and present this data using pictographs and simple bar graphs.

Working mathematically

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.

Structure

In Mathematics, standards for the Structure dimension are introduced at Level 3.

Year 3 National Numeracy Benchmarks

The benchmarks describe minimum standards. For this reason, the Year 3 benchmarks relate to Level 2 Mathematics standards. Numeracy benchmarks are located at Curriculum Corporation.

Learning focus

As students work towards the achievement of Level 3 standards in Mathematics, they recognise and explore patterns in numbers and shape. They increasingly use mathematical terms and symbols to describe computations, measurements and characteristics of objects.

In Number, students use structured materials to explore place value and order of numbers to tens of thousands. They skip count to create number patterns. They use materials to develop concepts of decimals to hundredths. They use suitable fraction material to develop concepts of equivalent fraction and to compare fraction sizes. They apply number skills to everyday contexts such as shopping. They extend addition and subtraction computations to three digit numbers. They learn to multiply and divide by single digit numbers.

In Space, students sort lines, shapes and solids according to key features. They use nets to create three-dimensional shapes and explore them by counting edges, faces and vertices. They visualise and draw simple solids as they appear from different positions. They investigate simple transformations (reflections, slides and turns) to create tessellations and designs. They explore the concept of angle as turn (for example, using clock hands) and as parts of shapes and objects (for example, at the vertices of polygons). They use grid references (for example, A5 on a street directory) to specify location and compass bearings to describe directions. They use local and larger-scale maps to locate places and describe suitable routes between them.

In Measurement, chance and data, students measure the attributes of everyday objects and events using formal (for example, metres and centimetres) and informal units(for example, pencil lengths). Students tell the time using analogue and digital clocks and relate familiar activities to the calendar. Students investigate natural variability in chance events and order them from least likely to most likely. Students conduct experiments and collect data to construct simple frequency graphs. They use simple two-way tables (karnaugh maps) to sort non-numerical data.

In Structure, students use structured material (in tens, hundreds and thousands) to develop ideas about multiplication by replication and division by sharing. They recognise the possibility of remainders when dividing. They learn to use number properties to support computations (for example, they use the commutative and associative properties for adding or multiplying three numbers in any order or combination). They investigate the distributive property to develop methods of multiplication and division by single digit whole numbers. They learn to use and describe simple algorithms for computations. They use simple rules to generate number patterns (for example, ‘the next term in the sequence is two more than the previous term’). They create and complete number sentences using whole numbers, decimals and fractions.

When Working mathematically, students use mathematical symbols (for example, brackets, division and inequality, the words and, or and not). Students develop and test ideas (conjectures) across the content of mathematical experience. For example:

• in Number, the size and type of numbers resulting from computations
• in Space, the effects of transformations of shapes
• in Measurement, chance and data, the outcomes of random experiments and inferences from collected samples.

Students learn to recognise practical applications of mathematics in daily life, including shopping, travel and time of day. They identify the mathematical nature of problems for investigation. They choose and use learned facts, procedures and strategies to find solutions. They use a range of tools for mathematical work, including calculators, computer drawing packages and measuring tools.

National Statements of Learning

This learning focus statement, with the following elaboration, incorporates the Year 3 National Statement of Learning for Mathematics.

Elaboration:

They recognise angles … as parts of shapes and objects …

Standards

Number

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 3/4 > 2/3 using physical models. 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).

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.

Space

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. 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. 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.

Measurement, chance and data

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

Structure

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.

Working mathematically

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.

Year 5 National Numeracy Benchmarks

The benchmarks describe minimum standards. For this reason, the Year 5 benchmarks relate to Level 3 Mathematics standards. Numeracy benchmarks are located at Curriculum Corporation.

Learning focus

As students work towards the achievement of Level 4 standards in Mathematics, they describe their investigations with correct mathematical terms, symbols and notations. They use mathematical procedures to construct and systematically investigate conjecture or hypotheses.

In Number, students extend their understanding of whole numbers, fractions and decimals. They use patterns and arrays to develop understanding of multiples (including lowest common multiple), factors (including highest common factor), prime and composite numbers. They recognise and use simple powers (for example, 2³ = 8).

Students investigate and use equivalent forms of common fractions. They order fractions and decimals and locate them on a number line. They investigate temperature and other contexts to develop the concept of negative numbers. They explore ideas of ratio (as a comparison) and percentage (comparing to 100). They use materials to explore decimals, ratios and percentages as equivalent forms of fractions (for example, 1/2 = 0.5 = 50% = 1 : 2).

Students devise and use mental and written methods (algorithms) to add, subtract, multiply and divide whole numbers. For division they recognise remainders as common fractions or decimals. They devise and use mental and written methods to add and subtract decimals. They use materials and number lines to develop understanding of multiplication and division of decimals (to two decimal places) and simple common fractions. They routinely make estimations and approximations in calculations and make judgments about their accuracy.

In Space, students identify and sort shapes by properties such as parallel and perpendicular lines (for example, quadrilaterals). They use the ideas of angle, size and scale to describe the features of shapes and solids. They identify symmetry by reflection or rotation. They create and compare pairs of enlarged shapes using simple scale factors. They describe the features that change (for example, side lengths) and features that remain the same (for example, angles). They represent solids (for example, prisms, pyramids, cylinders and cones) as two-dimensional drawings and nets. They visualise and describe relative location and routes between places shown on a map. They create and interpret simple networks such as a road network to show connectedness between towns.

In Measurement, chance and data, students estimate and measure lengths (including perimeter), area (including surface area), volumes, capacity, time (including duration), and temperature in metric units using appropriate instruments and scales. They determine and use the level of accuracy required for the purpose of the measurement. They develop simple procedures to determine the perimeter and area of simple shapes (for example, counting squares in a grid to determine area).

Students estimate and describe the chance of random events using words, percentages and fractions or decimals between 0 and 1. They investigate the sample space (possible outcomes) for simple chance events and calculate theoretical probability. They explain how symmetry in chance situations (for example, the roll of a die) creates equally likely outcomes. They create simulations of chance events to estimate probability (for example, randomly selecting a card from a pack without kings to choose a month).

Students plan and conduct questionnaires to collect data for a specific purpose. They recognise different data types such as categorical and numerical, discrete and continuous. They organise and present grouped and ungrouped data using displays such as simple frequency tables and histograms. They calculate and interpret measures of centre (mean, median and mode) and spread (range) for ungrouped data.

In Structure, students use venn diagrams and tables (karnaugh maps) to test the validity of statements involving the quantifiers none, some and all. They develop algorithms involving words, diagrams and mathematical symbols (for example, for testing the divisibility of a number).

Students create number sequences by computing the next term from the previous term or terms (recursion). They develop function rules for the terms in sequences based on their position in the sequence.

Students recognise that the ‘identity’ for each operation has no effect: the number 0 for addition and subtraction, and 1 for multiplication and division. They form and solve equations using words and symbols.

When Working mathematically, students make and test conjectures and generalisations about numbers, shapes and mathematical structure using concrete materials and diagrams. For example:

• in Number, the factors of primes and composites
• in Space, the properties of shapes
• in Measurement, chance and data, the probability of outcomes in games of chance
• in Structure, the patterns of remainders formed by division.

Students identify and investigate real life, practical and historical applications of mathematics. They pose and solve mathematical problems using a range of strategies (for example, make a list, find a pattern, work backwards). They solve new problems based on familiar problem structures.

Students develop and use estimation procedures to check the results of computations made using technology. They use technology for complex and extended computations. They use appropriate technology to explore puzzles involving numbers (for example, solve a magic square using a spreadsheet) and to generate drawings of shapes, solids, nets and geometric designs.

National Statements of Learning

This learning focus statement incorporates the Year 5 National Statement of Learning for Mathematics.

Standards

Number

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).

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

Space

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. 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).

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.

Measurement, chance and data

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).

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.

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). 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).

Structure

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.

Working mathematically

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.

Year 7 National Numeracy Benchmarks

The benchmarks describe minimum standards. For this reason, the Year 7 benchmarks relate to Level 4 Mathematics standards. Numeracy benchmarks are located at Curriculum Corporation.

Learning focus

As students work towards the achievement of Level 5 standards in Mathematics, they construct mathematical models to explore and describe the physical world. They recognise the importance of mathematics in a technological society.

In Number, students investigate and explore whole numbers and fractions as squares, square roots and other simple powers. They express natural numbers as products of prime number factors.

Students use number lines and materials to compare quantities using ratios, and to form equal ratios using proportion. They use ratios of number pairs to understand constant rate of change. They use number lines, graphs, numerical or algebraic means to solve proportion problems and percentage problems as proportion relative to 100.

Students use patterns with division to develop understanding of infinite decimals, and recognise the existence and applications of non-repeating infinite decimals (for example, π). Students use mental, written or calculator methods for computations, including multiple operations using rounding and estimation to provide suitable answers for practical situations. They use materials and patterns to understand binary numbers and to add and subtract using this notation.

In Space, students construct shapes and regular polygons to given specifications. They explore the properties of angles formed by intersecting straight lines. They use ideas of congruency and similarity to create and describe designs and tessellations. They use nets and isometric diagrams for common three-dimensional shapes to construct corresponding geometric objects. They use perspective to draw three-dimensional objects on paper.

Students interpret and use a range of familiar and common maps of locations from small to large scale, using plans and grids. They explore the patterns formed by following procedures involving simple transformations or movements around grids. They use networks to represent relationships in everyday life (for example, a tree diagram for a family tree and a network to show the route used to travel to school).

In Measurement, chance and data, students use metric units to estimate and measure length, perimeter, area, surface area, mass, volume, capacity, angle in shapes and solids, time, and temperature. They convert metric units into smaller or larger units as required. They judge the accuracy of their estimates by measurement and calculate error. They use mensuration formulas (for example, for area and perimeter of circles, area and perimeter of triangles and parallelograms, and the surface area and volume of prisms and cylinders). They solve problems involving simple rates (per unit time or area).

Students estimate probability from simulations involving generation of random numbers and data of long-run frequencies. They calculate theoretical probabilities involving one- and two-event trials.

Students take samples in order to make inferences and predictions about a population. They learn to present data in appropriate graphical formats. They calculate and interpret summary statistics (mean, median, mode and range).

In Structure, students use diagrams to show the relationships between natural, integer, rational and irrational numbers. They give examples of the use of number properties (commutative, associative and distributive) and use counter-examples to show where they do not apply. They test logical equivalence of sentences using the quantifiers none, some and all and set operations of complement, intersection and union, by means of diagrams.

Students use the opposite of any integer for addition, and the inverse of any rational number for multiplication (reciprocal) to rearrange formulas and simple algebraic expressions and to solve equations. They use linear and other simple functions of a single variable, to explore number patterns and provide models for practical situations. They represent functions by tables of values, ordered pairs, graphs and rules applied over a given domain. They solve equations and inequalities with a sequence of inverse operations.

When Working mathematically, students determine different but equivalent ways to describe a set, using attributes linked by and, or, not, and by ideas of implication and equivalence. They generalise from multiple examples and informally justify those generalisations. They use linear and other simple mathematical models to explore practical situations. They make and test predictions from these models (including interpolation and extrapolation). They use technologies such as geometry software, graphics calculators and spreadsheets.

National Statements of Learning

This learning focus statement, with the following elaboration, incorporates the Year 7 National Statement of Learning for Mathematics.

Elaboration:

They construct three-dimensional objects from … isometric diagrams.

Standards

Number

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).

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.

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).

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%).

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.

Space

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. 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.

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.

Measurement, chance and data

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.

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

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.

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.

Structure

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 = 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).

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.

Working mathematically

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.

Learning focus

As students work towards the achievement of Level 6 standards in Mathematics, they extend their use of mathematical models to a wide range of familiar and unfamiliar contexts. They recognise the role of logical argument and proof in establishing mathematical propositions.

In Number, students investigate familiar and unfamiliar situations and contexts involving the use of all types of real numbers. They use irrational numbers such as φ , π, and common surds in calculations in both exact and approximate form. They apply mental, written or technology-assisted forms of computation as appropriate, using estimation to validate their answers. They compute using large or small numbers expressed in scientific notation. They evaluate and use factorials in relevant contexts. They apply the concepts of rounding to either a given number of decimal places or significant figures to check the accuracy of computations.

In Space, students investigate the possible orientation of lines in space. They investigate the properties of angles formed when lines (including parallel lines) intersect. They learn how space is enclosed in two and three dimensions, and systematically investigate the properties of boundaries and regions on surfaces with shapes such as polygons and circles, prisms and polyhedra (including the platonic solids). They learn to use the concepts of congruency and similarity to compare the size and shape of polygons. They investigate the properties of similar triangles.

Students investigate the relationship between position, length and angle using the pythagorean relationship and trigonometry of right-angled triangles. They explore simple combinations of rotations, translations and reflections as transformations of geometric shapes in the plane. They investigate the paths (loci) formed by points, lines and shapes as they move in space according to various rules, conditions and/or constraints involving transformations. They use symmetry and other properties to create tessellations in two and three dimensions from regular and composite shapes. They investigate the effects of changing the scale of one characteristic of a geometric shape (for example, length or angle) on the size of related characteristics (for example, area and volume).

Students use maps and globes to investigate location and distances between places.

In Measurement, chance and data, students measure and estimate perimeter, area, surface area, mass, volume, capacity, angle, and the rates of speed, density and concentration. They use and convert units to suit the purpose of the measurements. They make judgments about errors in measurement. They use formulas (including trigonometry) to calculate perimeters, areas, angles in shapes, and the surface areas and volumes of solids. They use degrees and radians, as applicable, for units of measurement of angles.

Students apply probability concepts to aspects of chance and risk in everyday life. They represent event spaces that show the nature of events and their probabilities, and use these representations to assist in the computation of the probabilities of compound, independent and dependent events. They apply the concept of mathematical expectation to describe expected gain or loss in games of chance.

Students collect and use uni-variate and bi-variate data samples. They select appropriate representations to display data distributions, centrality, spread, and association between bi-variate data sets.

In Structure, students learn to categorise natural, integer, rational and irrational numbers in relation to real numbers. They use the concepts of order, discrete and continuous, and finite and infinite in relation to these sets of numbers.

Students apply algebraic properties (for example, closure, associative, commutative, identity, inverse and distributive) to expressions, formulas and equations.

They relate sets with one, two or three attributes, in four ways:

• diagrams and grids
• the logical connectives and, or, not, implication and equivalence
• the quantifiers none, some and all
• the set operations complement, intersection, union and inclusion.

Students work with functions (for example, linear, quadratic, reciprocal, exponential), simple transformations of these functions, their graphs and related algebraic properties. They solve equations of the form f(x) = k, where k is a real constant. They solve simultaneous linear equations using algebraic, numerical and graphical approaches.

When Working mathematically, students develop generalisations by abstracting the features from situations, expressing these in words and symbols. They test propositions, and use formal mathematical arguments to test their truth, modifying them as required.

Students choose, use and develop mathematical models and procedures with attention to assumptions and constraints (for example, they test the suitability of the results of data analysis in terms of the context being modelled).

They solve problems in a wide range of practical, theoretical and historical contexts and communicate the results of these investigations. They extend their problem solutions by generalising, or changing the initial constraints of a situation for further investigation.

Students use technology (for example, geometry software, graphics calculators, spreadsheets and computer algebra systems) to develop mathematical ideas and solve problems.

They describe the major features of mathematical structure, and use of logical argument in mathematical discourse and applications of mathematics.

National Statements of Learning

This learning focus statement incorporates the Year 9 National Statement of Learning for Mathematics.

Standards

Number

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).

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).

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.

Space

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. They recognise the features of circles (centre, radius, diameter, chord, arc, semi-circle, circumference, segment, sector and tangent) and use associated angle properties.

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 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).

Measurement, chance and data

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.

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

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.

Structure

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).

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.

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.

Working mathematically

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.