Mathematics Level 6 (Years 9 and 10)
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, with the following elaboration, 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) = 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 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.
Downloads
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Mathematics booklet (
PDF - 682KB) - Mathematics standards table (Doc - 118KB)
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Victorian Essential Learning Standards Level 6 ( PDF - 788KB)
