Mathematics Level 5 (Years 7 and 8)
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 = 2x 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.
Downloads
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Mathematics booklet (
PDF - 682KB) - Mathematics standards table (Doc - 118KB)
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Victorian Essential Learning Standards Level 5 ( PDF - 755KB)
