LEARNING+STAGES

**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. **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. **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 101 2 × 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. **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, = 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 f, and simple fractions of p 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 9 (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 p (for example p + p/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 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. **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. **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 **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). **__MC&D__** **//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). **//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). **//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. **//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 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. **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. **At Level 5** students identify collections of numbers as subsets of natural numbers, integers, rational numbers and real numbers. They use Venn diagrams and tree diagrams to show the relationships of intersection, union, inclusion (subset) and complement between the sets. They list the elements of the set of all subsets (power set) of a given finite set and comprehend the partial-order relationship between these subsets with respect to inclusion (for example, given the set {a, b, c} the corresponding power set is {Ø, {a}, {b}, {c }, {a, b }, {b, c}, {a, c }, {a, b, c}}.) They test the validity of statements formed by the use of the connectives and, or, not, and the quantifiers none, some and all, (for example, ‘some natural numbers can be expressed as the sum of two squares’). They apply these to the specification of sets defined in terms of one or two attributes, and to searches in data-bases. Students apply the commutative, associative, and distributive properties in mental and written computation (for example, 24 × 60 can be calculated as 20 × 60 + 4 × 60 or as 12 × 12 × 10). They use exponent laws for multiplication and division of power terms (for example 23 × 25 = 2 8, 20 = 1, 23 ÷ 25 = 2-2, (52)3 = 56 and (3 × 4)2 = 3 2 × 42). Students generalise from perfect square and difference of two square number patterns (for example, 252 = (20 + 5)2 = 400 + 2 × (100) + 25 = 625. And 35 × 25 = (30 + 5) (30 - 5) = 900 - 25 = 875) Students recognise and apply simple geometric transformations of the plane such as translation, reflection, rotation and dilation and combinations of the above, including their inverses. They identify the identity element and inverse of rational numbers for the operations of addition and multiplication (for example, ½+ - ½ = 0 and 2 / 3 × 3 /2 = 1). Students use inverses to rearrange simple mensuration formulas, and to find equivalent algebraic expressions (for example, if P = 2L + 2W, then W = P / 2 - L . If A = p r2 then r = vA /p). They solve simple equations (for example, 5 x + 7 = 23, 1.4x - 1.6 = 8.3, and 4x 2 - 3 = 13) using tables, graphs and inverse operations. They recognise and use inequality symbols. They solve simple inequalities such as y = 2x+ 4 and decide whether inequalities such as x 2 > 2y are satisfied or not for specific values of x and y. Students identify a function as a one-to-one correspondence or a many-to-one correspondence between two sets. They represent a function by a table of values, a graph, and by a rule. They describe and specify the independent variable of a function and its domain, and the dependent variable and its range. They construct tables of values and graphs for linear functions. They use linear and other functions such as f( x) = 2x - 4, xy = 24, y = 2x and y = x2 - 3 to model various situations. **At Level 6**, students classify and describe the properties of the real number system and the subsets of rational and irrational numbers. They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations. Student express relations between sets using membership, ?, complement, ', intersection, n, union, ?, and subset, ?, for up to three sets. They represent a universal set as the disjoint union of intersections of up to three sets and their complements, and illustrate this using a tree diagram, venn diagram or karnaugh map. Students form and test mathematical conjectures; for example, ‘What relationship holds between the lengths of the three sides of a triangle?’ They use irrational numbers such as, p, f and common surds in calculations in both exact and approximate form. Students apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to computation with number, to rearrange formulas, rearrange and simplify algebraic expressions involving real variables. They verify the equivalence or otherwise of algebraic expressions (linear, square, cube, exponent, and reciprocal, (for example, 4 x - 8 = 2(2x - 4) = 4(x - 2); (2a - 3)2 = 4a2 - 12a + 9; (3w)3 = 27w3 ; (x3y) / xy2 = x2y -1; 4 /x y = 2/x × 2/y).  Students identify and represent linear, quadratic and exponential functions by table, rule and graph (all four quadrants of the Cartesian coordinate system) with consideration of independent and dependent variables, domain and range. They distinguish between these types of functions by testing for constant first difference, constant second difference or constant ratio between consecutive terms (for example, to distinguish between the functions described by the sets of ordered pairs {(1, 2), (2, 4), (3, 6), (4, 8) …}; {(1, 2), (2, 4), (3, 8), (4, 14) …}; and {(1, 2), (2, 4), (3, 8), (4, 16) …}). They use and interpret the functions in modelling a range of contexts. They recognise and explain the roles of the relevant constants in the relationships f(x) = a x + c, with reference to gradient and y axis intercept, f(x) = a (x + b)2 + c and f(x ) = cax.  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 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. **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. **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. **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.
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