Level 6
Number and Algebra
Number strategies and knowledge
AO1: Apply direct and inverse relationships with linear proportions. This means students will solve problems that linear proportions. The term "Linear proportions" encompasses a broad range of contexts including rates and ratios, scaling, probability, conversion between measures, derived measures such as density, conversion between numbers forms, partitioning and replicating, and using rational numbers in operations. Proportional thinking is applied across the objectives of all strands at Level Six. Structurally, problems involving direct linear proportions are of three main types:
- Find a missing value in an equality of the form a/b = c/d where one of the values a,b,c, or d is unknown, e.g. "Find 65% of 43", can be represented as 65/100 = x/43.
- Determine the size relationship between a/b and c/d, e.g. "which is the stronger concentration of syrup to water; 2:7 or 3:11?" can be represented as 2/9 > 3/14 (the fractions represent the part to whole relationships).
- Find values for a, b, c, or d that satisfy the inequality a/b< c/d, e.g. For what positive integer values of x is the following inequality true, 2/5 > 6/x?
Inverse relationships in this objective refer to two types:
- Apply an inverse operation where the given information requires it, e.g. "36% of what amount is $26.64?" can be represented as 36x/100 = 26.64.
- Solve problems with inverse proportions, e.g. "A car travels from A to B in 25 minutes at 100 kilometres per hour. How long will the trip take at 80 kilometres per hour?" can be represented as 25 x 100 = 80x.
- Both direct and inverse proportional relationships should be represented through equations (as above), tables (including spreadsheets) and graphs.
AO2: Extend powers to include integers and fractions. This means students will extend their understanding or powers to include powers involving fractions and integers. The conventions for the meaning of negative and fractional exponents are derived from the preservation of the number laws for exponents, ab x a = ab + c and ab ÷ ac = ab - c, e.g. 42 x 43 = 45 so 45 ÷ 42 = 43. So the meaning of a-b must preserve the truth of a-b x ab = ab + -b. Since ab + -b = a0 = 1, a-b must be the reciprocal of ab ( 1/ab) since ab x 1/ab = 1. For example, 63 = 216 so 6-3 = 1/63 = 1/216. The meaning of a1/b must preserve the truth of (a1/b)b = 1, e.g. a1/2x a1/2 = a1 = a, so a1/2 must be the square root of a (√a) and a1/3x a1/3 x a1/3 = a11 = a, so a1/3 must be the cube root of a (3√a). So in general a1/b = b√a. At Level Six students should accept that negative and fractional exponents behave in the same way as positive integral exponents and use the number laws to solve problems, e.g. If 43 = 64 and √4 = 2, what is 43 1/2? Or If 3√8 = 2, what is 8 2/3? .
AO3: Apply everyday compounding rates. This means students will solve problems that involve examples of everyday compounding rates. "Compounding rates" refers to situations in which a quantity is compounding, over time, as a fixed rate is applied cumulatively. The most common example of this is compound interest on bank deposits. In this situation the interest earned in one year becomes part of the total amount on which interest is calculated for the following year. Other examples include simple situations of growth and decay, e.g. inflation, bacterial growth, half-lives..
AO4: Find optimal solutions, using numerical approaches.
This means students will structure calculations to find optimal solutions. Optimal solutions are
those that maximise or minimise a quantity of importance while meeting the constraints of a situation.
For example, "find the square based prism with a volume of 700cm3 that has the minimum surface area", involves
minimising surface area while meeting the constraints of given shape and volume. Numerical approaches involve
structuring calculations in a systematic way so that the optimal solution is found, and considering the degree of
accuracy required. Usually this involves constructing a table (spreadsheet is an example). For example:
At Level Six students should be able to:
- Construct a table that contains the relevant variables without unnecessary duplication, e.g. Edges A and B are defined by the same measure in this problem.
- Create formulae that calculate the quantities required by the problem, e.g. Length of Edge C and surface area in this problem.
- Recognise and act on the need to work within values initially selected to increase the level of accuracy, e.g. work with decimals values for Edge A and B between 8 and 10 for this problem
Equations and expressions
AO1: Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns. This means students will create equations to model everyday situations, e.g. express a taxi charge as a linear equation (flagfall and kilometre rate) or the exponential relationship between the number of repeated folds (in thirds) of a paper strip and the number of sections formed. This includes forming pairs of simultaneous linear equations. Students should be able to form equations from tables of values, using differences between terms, constant first order for linear relations, constant second order differences for quadratic relations and constant ratio for simple exponentials. They should use algebraic manipulation skills to simplify expressions, including rational expressions involving exponents, e.g. 9n4 / 6n3. Students should apply their manipulations skills to solve linear and quadratic equations by applying inverse operations with an appreciation of equality and connect their solutions to corresponding situations of inequality, e.g. If (6x - 8)/4 = 10 has the solution x = 8 then (6x - 8) /4 < 10 has the solution x < 8. They should be able to solve quadratic equations by factorising and have the disposition and capability to check all of their algebraic solutions by substituting values. Solving simple exponential equations should be done by inspection at this level, e.g. 3x = 81 by recognising 34 = 81 so x = 4. Pairs of simultaneous equations may be solved by substitution, elimination and by intercept of graphs.
Patterns and relationships
AO1: Generalise the properties of operations with rational numbers, including the properties of exponents. This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to examine a number of cases, define the variables involved, use appropriate mathematical terminology and symbols, and ultimately reason with the properties themselves. Rational numbers are defined as those that can be expressed in the form a/b, where a and b are integers and b ? 0. At Level Six students should be able to describe and apply the properties of addition, subtraction, multiplication and division as these operations apply to rational numbers and exponents. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express the generalisations algebraically, e.g. the commutative property of addition may be represented by a/b + c/d = c/d + a/b. Students should be able to express the multiplication and division of exponents with common bases algebraically, e.g. (an)/ am = an - m, and derive other properties of exponents by applying first principles to specific cases, e.g. 32 x 42 = 3 x 3 x 4 x 4 = 12 x 12 = 122 leading to anbn = (ab)n .
AO2: Relate graphs, tables, and equations to linear, quadratic, and simple exponential relationships found in number and spatial patterns. This means students will relate the shape of graphs to the type of relation they show; lines for linear, parabolas for quadratics and exponential curves. They should also be able to find the equations using the slope and intercept for linear equations (relating slope to the constant between y-values), and using vertex, orientation and specific ordered pairs of parabolas for quadratics. They should recognise the relationship involved considering the difference and ratios between terms in tabular form. The spatial and number patterns involved are those that yield appropriate co-variation between variables within the pattern. The advantage of spatial patterns is that students are able to generate potential relationships by attending to the spatial elements within the pattern and validate their findings through mapping back to the pattern itself. Students at Level Six should connect the spatial, tabular, graphic and equation representations of a relationship and choose which representation they see as most useful to solve a given problem. For example, consider the growing diamond pattern:
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Year One |
Year Two |
Year Three |
The relationship between years and matches could be expressed as:
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...or as an equation m = 2y2 + 2y where m represents the numbers of matches and y the number of years.
AO3: Relate rate of change to the gradient of a graph. This means students will connect the difference between successive terms in a relationship as the rate of change and know how this shows in graphical representation. They should do so through interpreting everyday contexts such as the speed of falling objects (e.g. parachutists), growth of organisms (e.g. algae) or compound growth (e.g. debt if unpaid). This includes knowing that constant differences between terms result in linear graphs and equations, constant second order differences and ratio suggest different models of varying rate of change that can be presented as quadratic or exponential equations. Students should be able to map from a graph to the situation that produced the graph, e.g. describe the speed of a car from a time and distance graph.
Geometry and Measurement
Measurement
AO1: Measure at a level of precision appropriate to the task.
This means students will be able to identify and use a unit of measurement the meets the requirements of a given task, for example, the length of a fence paling in millimetres. This involves understanding of the accuracy required in a given context including appreciation of the practical consequences of both less accuracy and greater precision. This includes the effect of using different measures of length on the accuracy of the resulting area and volume measures. Students need to understand the necessary compromise at times between accuracy and adequacy, that is, something could be measured more precisely but there is greater effort required and no gain to meeting the demands of the task. This objective also includes making sensible estimates where appropriate, for example, estimating the number of rolls of wallpaper or litres of paint needed for a given space.
AO2: Apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures.
This means students will know the commonly used units including the role of prefixes as conversion factors of base units, e.g. kilo meaning one thousand, micro meaning one millionth. Below is a list of units for key attributes that should be expected.
Attribute | Units |
Length | metre (m), micrometre ( m), millimetre (mm), centimetre (cm), kilometre (km) |
Area | square metre (m2), square millimetre (mm2),square centimetre (cm2), hectare (ha), square kilometre (km2) |
Volume | cubic metre (m3), cubic centimetre (cm3), cubic decimetre (litre),cubic kilometre (km3) |
Capacity | litre (L), millilitre (mL), decilitre (dL) |
Mass | gram (g), microgram (μ g), milligram (mg), kilogram (kg), tonne (t) |
Time | second (s), microsecond (μs), millisecond (ms), minute, hour, day, etc | Temperature | degree Celsius (° C). |
Angle | degree (° ). |
Students are also expected to know derived measures that describe rates involving the units above and other common units. Attributes and the derived units used to measure them include speed (kilometres per hour km/h, metres per second m/s), fuel and energy consumption (litres per 100 kilometres L/100km, joules or calories per minute ), unit price (cents or dollars per gram), and density (kilograms per cubic metre kg/m3 , grams per cubic centimetre g/cm3). More complicated derived measures such as those for pressure, force, and power are not expected at Level Six.
Students should be able to connect the units for volume (capacity) and mass, e.g. Find the mass of 345mL of water, and convert between the simple derived units above, e.g. 140 km/h = 
m/s.
AO3: Calculate volumes, including prisms, pyramids, cones, and spheres, using formulae.
This means students will connect the formulae for the volume of prisms, including cylinders, as area of the cross-section or base multiplied by the third dimension, e.g. for a rectangular based prism v = l x w x h (l x w is the area of the cross section). This involves recognising how to apply the formula for prisms given any orientation of the solid that is presented. Similarly students should connect the formulae for the volume of pyramids, including cones, as the area of the base multiplied by one third of the height, e.g. for a cone v = πr2h (πr2 is the area of the base). Students should know and apply the formula for volume of a sphere as v = 4/3 πr3. At Level Six students are expected to work with decimal measures as well as whole number measures.
Shape
AO1: Deduce and apply the angle properties related to circles.
This means students will know and apply the sum of interior angles of a triangle (180°) and the angle between a radius and tangent (90°) to deduce the angle properties related to circles.
The angle properties of circles expected are:
- Angle at centre to any chord is twice the angle at circumference.
- Angles at the circumference to any chord are equal. .
- Angle between a chord and a tangent equals the angle in the opposite segment. .
- For a triangle in a semi-circle the angle at circumference equals 90?..
- Opposite angles in a cyclic quadrilateral add to 360?. .
Students are expected to connect at least two of these properties to find unknown angles in a given problem and to communicate their reasoning, citing the angle properties used.
AO2: Recognise when shapes are similar and use proportional reasoning to find an unknown length.
This means students will know what properties of shapes are conserved as they are enlarged (or reduced) to scale. In particular this refers to angles and the ratios of side lengths within a figure and between a figure and its enlargement. They should also apply knowledge that area increases by the square of the scale factor and volume increases by the cube of the scale factor. Students should solve problems in which they find unknown lengths of shapes that are both regular and irregular. For example: Given that the ellipses are similar, find the unknown length of the major axis.
Given the rectangles are similar find the values of c and d.
AO3: Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions.
This means students will , given the required measurements, be able to connect trigonometric ratios (sine, cosine or tangent) in two dimensions with the demands of three dimensional problems, usually applying Pythagoras’ theorem in two or three dimensions in doing so. The problems should involve finding either an unknown length or angle. For example, find the angle <abc and the length of the line bc.
This objective also applies to problems where points are described using co-ordinates in two dimensions (four quadrants). For example, a triangle has corners at (2,3), (1,7), and (5,5). Find the lengths of its sides.
Position and orientation
AO1: Use a co-ordinate plane or map to show points in common and areas contained by two or more loci.
This means the students will be able to use algebra and graphing to find a point in common with two intersecting lines when given their equations and connect this understanding with contexts that can be modelled with simultaneous linear equations in two variables. A loci is a set of points satisfying a given condition so common examples are lines, circles and ellipses, parabolas, and hyperbolas.
They should be able to sketch the locus for a given condition and recognise when that condition meets that of a conic section, e.g. perimeter of the flight area of a jet taking off and returning to a moving aircraft carrier (ellipse). Using graphing techniques students should be able to find the point or points in common between a line (given two points) and a conic (given the condition) and describe the area bounded by common lines and conics, for example, the grazing area of a tethered animal constrained by a wall.
Transformation
AO1: Compare and apply single and multiple transformations.
This means students will be able to draw, with the assistance of technology where available, the results of transformations acting successively on a figure, for example, frieze patterns. The transformations involved are reflection, rotation, translation and enlargement. They should recognise when combinations of transformations give the same or a different result, for example, reflection then translation has the same result as translation then reflection (glide reflections), and acknowledge this in describing which transformations result in a figure being mapped onto a given image. Students should also connect the result of translations and reflections on lines and parabolas with the similarities and differences in their equations, for example, the image of y = x2 + 3 reflected in the x-axis is y = - (x2 + 3) or y = - x2- 3.
AO2: Analyse symmetrical patterns by the transformations used to create them.
This means students will apply their knowledge of variant and invariant properties under these translations in explaining how they determined which translations were involved in a given mapping. This includes attendance to equality of lengths and angles, and order (direction). For example, students should describe how the following frieze pattern may have been created from the arrow element... (possibilities include)
Statistics
Statistical investigation
AO1: Plan and conduct investigations using the statistical enquiry cycle:
Further detail on this Achievement Objective will be added shortly.
Statistical literacy
AO1: Evaluate statistical reports in the media by relating the displays, statistics, processes, and probabilities used to the claims made. Further detail on this Achievement Objective will be added shortly.
Probability
AO1: Investigate situations that involve elements of chance:
Further detail on this Achievement Objective will be added shortly.
