NSW Syllabuses

# Stage Statements

Stage statements are summaries of the knowledge, skills, understanding, values and attitudes that have been developed by students as a result of achieving the outcomes for each stage of learning.

### Prior-to-school learning

Students bring to school a range of knowledge, skills and understanding developed in home and prior-to-school settings. The movement into Early Stage 1 should be seen as a continuum of learning and planned for appropriately.

The Early Years Learning Framework for Australia describes a range of opportunities for students to learn and develop a foundation for future success in learning.

The Early Years Learning Framework for Australia has five learning outcomes that reflect contemporary theories and research evidence concerning children's learning. The outcomes are used to guide planning and to assist all children to make progress.

The outcomes are:

1. Children have a strong sense of identity
2. Children are connected with and contribute to their world
3. Children have a strong sense of wellbeing
4. Children are confident and involved learners
5. Children are effective communicators.

In addition, teachers need to acknowledge the learning that children bring to school, and plan appropriate learning experiences that make connections with existing language and literacy development, including language used at home.

### Early Stage 1

By the end of Early Stage 1, students ask questions and use known facts to explore mathematical problems and develop fluency with mathematical ideas. They use everyday language, concrete materials and informal recordings to demonstrate understanding and link mathematical ideas.

Students count to 30 and represent numbers to 20 with objects, pictures, numerals and words. They read and use ordinal numbers to at least 'tenth'. Students use concrete materials to model addition, subtraction, multiplication and division. They use the language of money and recognise the coins and notes of the Australian monetary system. Students divide objects into two equal parts and describe them as halves. They recognise, describe and continue repeating patterns of objects and drawings.

Students identify length, area, volume, capacity and mass, and compare and arrange objects according to these attributes. They manipulate, sort and represent three-dimensional objects and describe them using everyday language. Students manipulate, sort and describe representations of two-dimensional shapes, identifying circles, squares, triangles and rectangles. They connect events and the days of the week and explain the order and duration of events, telling the time on the hour. Students give and follow simple directions and describe position using appropriate language.

Students answer simple questions to collect information. They use objects to create a data display and interpret data.

### Stage 1

By the end of Stage 1, students ask questions and use known facts, objects, diagrams and technology to explore mathematical problems and develop mathematical fluency. They link mathematical ideas and use appropriate language and diagrams to explain strategies used.

Students count, order, read and write two- and three-digit numbers and use a range of strategies and recording methods. They use mental strategies and concrete materials to add, subtract, multiply and divide, and solve problems. Students model and describe objects and collections divided into halves, quarters and eighths. They associate collections of Australian coins with their value. They use place value to partition numbers. Students describe and continue a variety of number patterns and build number relationships. They relate addition and subtraction facts for sums to at least 20.

Students estimate, measure, compare and record using informal units for length, area, volume, capacity and mass. They recognise the need for formal units of length and use the metre and centimetre to measure length and distance. They use a calendar to identify the date and name and order the months and the seasons of the year. Students use informal units to compare and order the duration of events and tell the time on the half- and quarter-hour. They identify, describe, sort and model particular three-dimensional objects and two-dimensional shapes. Students represent and describe the position of objects and interpret simple maps.

Students collect, organise, display and interpret data using lists, tables and picture graphs. They recognise and describe the element of chance in everyday events.

### Stage 2

By the end of Stage 2, students ask questions and use efficient mental and written strategies with increasing fluency to solve problems. They use technology to investigate mathematical concepts and check their solutions. Students use appropriate terminology to describe and link mathematical ideas, check statements for accuracy and explain their reasoning.

Students count, order, read and record numbers of up to five digits. They use informal and formal mental and written strategies to solve addition and subtraction problems. Students use mental strategies to recall multiplication facts up to 10 × 10 and related division facts. They use informal written strategies for multiplication and division of two-digit numbers by one-digit numbers. Students represent, model and compare commonly used fractions, and model, compare and represent decimals of up to two decimal places. Students perform simple calculations with money and solve simple purchasing problems. They record, describe and complete number patterns and determine missing numbers in number sentences. Students recognise the properties of odd and even numbers.

Students estimate, measure, compare, convert and record length, area, volume, capacity and mass using formal units. They read and record time in hours and minutes, convert between units of time, and solve simple problems involving the duration of time. Students name, describe and sketch particular three-dimensional objects and two-dimensional shapes. They combine and split two-dimensional shapes to create other shapes. They compare angles using informal means and classify angles according to their size. Students use a grid-reference system to describe position, and compass points to give and follow directions. They make simple calculations using scales on maps and plans.

Students collect and organise data, and create and interpret tables and picture and column graphs. They list all possible outcomes of everyday events, and describe and compare chance events in social and experimental contexts.

### Stage 3

By the end of Stage 3, students ask questions and undertake investigations, selecting appropriate technological applications and problem-solving strategies to demonstrate fluency in mathematical techniques. They use mathematical terminology and some conventions, and they give valid reasons when comparing and selecting from possible solutions, making connections with existing knowledge and understanding.

Students select and apply appropriate mental, written or calculator strategies for the four operations and check the reasonableness of answers using estimation. They solve word problems and apply the order of operations to number sentences where required. Students identify factors and multiples and recognise the properties of prime, composite, square and triangular numbers. They connect fractions, decimals and percentages as different representations of the same value. Students compare, order and perform calculations with simple fractions, decimals and percentages and apply the four operations to money in real-life situations. Students record, describe and continue geometric and number patterns, and they find missing numbers in number sentences. They locate an ordered pair in any one of the four quadrants on the Cartesian plane.

Students select and use the appropriate unit to estimate, measure and calculate length, area, volume, capacity and mass. They make connections between capacity and volume, and solve problems involving length and area. Students use 24-hour time in real-life situations, construct and interpret timelines and use timetables. They convert between units of length, units of capacity and units of mass. They construct and classify three-dimensional objects and two-dimensional shapes, and compare and describe their features, including line and rotational symmetries. Students measure and construct angles, and find unknown angles in diagrams using known angle results. They use a grid-reference system to locate landmarks and describe routes using landmarks and directional language.

Students use appropriate data collection methods to interpret and analyse sets of data and construct a range of data displays. They assign probabilities as fractions, decimals or percentages in simple chance experiments.

### Stage 4

By the end of Stage 4, students use mathematical terminology, algebraic notation, diagrams, text and tables to communicate mathematical ideas, and link concepts and processes within and between mathematical contexts. They apply their mathematical knowledge, skills and understanding in analysing real-life situations and in systematically exploring and solving problems using technology where appropriate. Students develop fluency with a range of algebraic techniques and in the solution of familiar problems. In solving particular problems, they compare the strengths and weaknesses of different strategies and solutions.

Students develop a range of mental strategies to enhance their computational skills. They operate competently with integers, fractions, decimals and percentages, and apply these in a range of practical contexts, including problems related to GST, discounts and profit and loss. Students are familiar with the concepts of ratios and rates, and apply these when solving problems. They investigate divisibility tests, use index notation for numbers with positive integral indices, and explore prime factorisation, squares and cubes, and related square and cube roots, and the concept of irrational numbers.

Extending and generalising number patterns leads students into an understanding of the use of pronumerals and the language of algebra. They simplify algebraic expressions, substitute into algebraic expressions and formulas, and expand and factorise algebraic expressions. Students solve simple linear and quadratic equations. They develop tables of values from linear relationships and illustrate these relationships on the Cartesian plane, with and without the use of digital technologies.

Students calculate the perimeters and areas of a variety of polygons, circles, sectors and simple composite figures, and solve related problems. They calculate the volumes and capacities of right prisms and cylinders, and solve related problems. They convert between units of area and units of volume, and connect units of volume and capacity. Pythagoras' theorem is used to calculate side lengths in right-angled triangles and solve problems in two dimensions. Students calculate time duration and apply their understanding of Australian and world time zones to solve problems.

Knowledge of the properties of two-dimensional geometrical figures, angles, parallel lines, perpendicular lines and congruent figures enables students to apply logical reasoning to solve numerical exercises involving unknown lengths and angles in figures.

Students construct, interpret and compare data displays, including dot plots, stem-and-leaf plots, sector graphs, divided bar graphs, and frequency tables and histograms. In analysing data, they consider both categorical and numerical (discrete and continuous) variables, sampling versus census, and possible misrepresentation of data, and calculate the mean, mode, median and range. Students represent events using Venn diagrams and two-way tables, and calculate the probability of simple and complementary events in single-step chance experiments.

### Stage 5.1

By the end of Stage 5.1, students explain and verify mathematical relationships, select and use appropriate strategies to solve problems, and link mathematical ideas to existing knowledge and understanding. They use mathematical language and notation to explain mathematical ideas, and interpret tables, diagrams and text in mathematical situations.

Students apply their knowledge of percentages, fractions and decimals to financial problems related to earning and spending money, taxation, and simple and compound interest. They simplify and evaluate numerical expressions using index laws for positive and zero indices, round numbers to a specified number of significant figures, and express numbers in scientific notation. Students apply the index laws to simplify algebraic expressions. They determine the midpoint, gradient and length of intervals on the Cartesian plane and draw graphs of linear and simple non-linear relationships.

Skills in measurement are further developed to include finding the areas of composite shapes and the surface areas of rectangular and triangular prisms. Students describe the limit of accuracy of measurements. They apply right-angled triangle trigonometry to practical situations, including those involving angles of elevation and depression. They apply the properties of similar figures to find side lengths in problems related to similar figures.

Students' statistical skills are extended to include considering shape and skewness of distributions, comparing data and data displays, and evaluating the reliability of statistical claims. They also determine the relative frequencies of events in chance experiments and calculate probabilities from information displayed in Venn diagrams and two-way tables.

### Stage 5.2

By the end of Stage 5.2, students use mathematical arguments to reach and justify conclusions. When communicating mathematical ideas, they use appropriate mathematical language and algebraic, statistical and other notations and conventions in written, oral or graphical form. Students use suitable problem-solving strategies, which include selecting and organising key information, and they extend their inquiries by identifying and working on related problems.

Students apply their knowledge of percentages, fractions and decimals to problems involving conversion of rates, direct proportion, and financial contexts related to compound interest and depreciation.

Students apply the index laws with integer indices to simplify expressions. They operate with algebraic fractions, expand binomial products and factorise monic quadratic trinomial expressions. They solve linear equations and use them to solve word problems. They solve linear inequalities and linear simultaneous equations. Students solve simple quadratic equations and solve monic quadratic equations by factorisation. On the Cartesian plane they draw and interpret graphs of straight lines, and simple parabolas, circles and exponential graphs. Students determine the equations of straight lines and use the properties of parallel and perpendicular lines on the Cartesian plane.

Students extend their skills in measurement to solve problems involving the surface areas and volumes of right prisms, cylinders and related composite solids. They use trigonometric ratios to solve problems in which angles may be measured to the nearest second, and problems involving bearings and angles of elevation and depression. In geometry, they use deductive reasoning in numerical and non-numerical problems, drawing on their knowledge of the properties of congruent triangles, the angle properties of polygons, and the properties of quadrilaterals.

Statistical skills are extended to include the construction of box-and-whisker plots and the calculation of interquartile range to analyse and compare data sets in appropriate data displays. Students investigate bivariate data sets and use scatter plots to describe relationships between variables. They evaluate the sources of data in statistical reports. In their study of probability, students record and determine probabilities of events in multi-step chance experiments and examine conditional language.

### Stage 5.3

By the end of Stage 5.3, students use deductive reasoning in problem solving and in presenting arguments and formal proofs. They interpret and apply formal definitions and generalisations and connect and apply mathematical ideas within and across substrands. They demonstrate fluency in selecting, combining and applying relevant knowledge, skills and understanding in the solution of familiar and unfamiliar problems.

Students operate with irrational numbers and extend their knowledge of the number system to include all real numbers. They analyse and describe physical phenomena and rates of change. Algebraic skills are extended to expanding the special binomial products and factorising non-monic quadratic expressions, using a variety of techniques. Students solve complex linear equations, non-monic quadratic equations, simple cubic equations, and simultaneous equations involving one linear and one non-linear equation. They solve practical problems using linear, quadratic and simultaneous equations. They change the subject of literal equations. Students generate, describe and graph straight lines, parabolas, cubics, hyperbolas and circles. They use formulas to calculate midpoint, gradient and distance on the Cartesian plane, and to determine the equations of straight lines.

Students solve problems involving the surface areas and volumes of pyramids, cones and spheres, and related composite solids. They explore similarity relationships for area and volume. They determine exact trigonometric ratios for 30°, 45° and 60°, extend trigonometric ratios to obtuse angles, and sketch sine and cosine curves for angular values from 0° to 360°. Students apply the sine and cosine rules for finding unknown angles and/or sides in non-right-angled triangles. They use Pythagoras' theorem and trigonometry to solve problems in three dimensions.

Their knowledge of a wide range of geometrical facts and relationships is used to prove general properties in geometry, extending the concepts of similarity and congruence to more generalised applications. Students prove known properties of triangles, quadrilaterals and circles.

Students use standard deviation to analyse data, and interpolate and extrapolate from bivariate data using lines of best fit. They investigate statistical reports and explore how data is used to inform decision-making processes.