Level 5
Number and Algebra
Number strategies and knowledge
AO1: Reason with linear proportions.
This means students will explore linear proportions in a variety of contexts. Linear proportions apply to situations which can be modelled using equivalent fractions, i.e. a/b = c/d where a,b,c, and d are integers (usually whole numbers). So proportional reasoning pervades many of the outcomes in all three strands and many contexts including:
- Using multiple ways to represent the same number in operator situations, for example 45% x 52 =
can be seen as 0.45 x 52 and 45/100 x 52, and comparing the potential results of operator situations by suspending calculation and thinking relationally, 30% x 34 = 60% x 17 (by doubling and halving) or 1.3 x 3.3 < 3.9 x 1.2 (since 1.3 x 3.3 = 3.9 x 1.1). - Comparing the results of sharing situations which involve fractional quotients, for example 3 pizzas shared among 5 boys (3/5 pizza each) results in a lesser share than 2 pizzas shared among 3 girls (2/3 pizza each), and find the difference in shares (2/3 -3/5=1/15) .
- Comparing the size of two fractions, decimals or percentages, using benchmark fractions or equivalence, give the difference between the fractions, and name a fraction between two fractions. For example, 4/7 > 5/9 since 4/7 is 1/14 greater than 1/2 and 5/9 is 1/18 greater than 1/2 or 4/7 = 36/63 and 5/9= 35/63 so the difference between 4/7and 5/9 is 1/63.
- Reasoning qualitatively about the size effect on a fraction as the numerator, denominator, or both numbers are changed. For example, given the fraction 5/11, reason that 5/10 and 6/11will be greater, 4/11 and 5/12 will be less, and comparing it with 4/10 and 6/12 will require further investigation.
- Measuring one fraction with another either by converting to equivalent forms or scaling the result of the same divisor acting on one. For example, each trip takes 3/4 of a full tank of petrol. You have 2/5 of a tank. What fraction of a trip can you make? as 3/4 = 15/20 and 2/5 = 8/20 so 8/20 is 8/15 of 15/20 (2/5 ÷3/4 = 8 /15) or 1 ÷3/4 = 4/3 (1 1/3 trips on a full tank) so 2/5 x 4/3 = 8/15 trips with two-fifths of a tank.
- Other examples of reasoning with linear proportions are discussed through the other achievement objectives.
AO2: Use prime numbers, common factors and multiples, and powers (including square roots).
This means students will know that prime numbers are numbers divisible by only themselves and one, and apply this to the fundamental law of arithmetic that every counting number has a unique prime factorisation, e.g. 36 = 2 x 2 x 3 x 3 = 22 32. They should apply prime factorisation to problems that involve factors and multiples, including finding the least common multiple or highest common factor. For example, “What sized cuboids can be made using 105 unit cubes?”, or “What is 105 out of 231 in simplest form?”
They should understand and use the additive law of exponents, that is ab x ac = ab+c and a b ÷ ac = a b - c and compare powers relationally (without calculation) where this is appropriate, e.g. 36 >63 because (3x3)x(3x3)x(3x3)>6x6x6. Students should understand the arithmetic and geometric origin of square roots. For example, a square of area 144cm2 has a side length of 12cm, and use common square roots to estimate the value of other square roots. For example, √36 = 6 and √49 = 7 so √42 ≈ 6.5. They should also understand the convention for negative exponents through pattern. For example 21= 2 so 20= 1 so 2 -1= 1/2 since the effect of decreasing the exponent by one is to divide the previous power by two.
AO3: Understand operations on fractions, decimals, percentages, and integers.
This means students will understand calculations involving fractions, decimals, percentages and integers assumes accuracy in calculation and the exercising of appropriate choice between mental, written and machine methods given the complexity of the numbers involved and the significance of the calculation in the context of the problem. Understanding also implied the prudent use of estimation to check the reasonableness of calculations and as an end in itself where approximations are sufficient.
Students should be able to explain the calculation steps (procedures) they followed and justify those steps by describing the quantities involved. For example, the calculation 1.4 x 0.6 = 0.84 might be justified as 14/10 x 6/10 = 84/100 or 14 x 6 = 84 and the size of answer being about half of 1.4.
The problems solved should involve result unknown, for example 56% of 38 = , change unknown, for example % of 38 = 21.28, or start unknown, for example 56% of = 21.28.
AO4: Use rates and ratios.
This means students will solve problems involving rates and ratios. In this curriculum rates are defined as a multiplicative relationship between different measures, for example, 24 litres per 60 minutes, while ratios are defined as a multiplicative relationship between identical measures, for example, 30 litres: 40 litres. This distinction is blurred where the measures are of the same attribute, for example, 10mL per 1 Litre, but problems involving unit conversion are delayed until Level Six. In terms of their behaviour problems involving both rates and ratios can be modelled by the equation a/b = c/d where one of the values, a, b, c, or d is unknown or as a situation where a/b and c/d must be compared. Rate and ratios can also be represented by ratio tables or double number lines. For example:
A wallpaper hanger mixes 300 grams of glue powder to every 4 litres of water. She wants to make up 25 litres of paste. How many grams of powder will she need?
At Level Five students are expected to solve problems of this type in which the unknown can be in any of the four positions on the table and in which the scalar within (e.g. 4 x = 25) or between (e.g. 4 x = 300) operators are positive integers or fractions. Students should be able to use equivalent rates to compare two given rates and express the part-whole relationships in ratios as equivalent fractions to compare given ratios. For example, 3 litre orange: 5 litres apple has a stronger orange flavour than 4:6 because the part-whole fractions are 3/8 and 4/10 respectively which have equivalent forms of 15/40 and 16/40.
AO5: Know commonly used fraction, decimal, and percentage conversions.
This means students will be able to express any of the fractions (halves, quarters, thirds, fifths, eighths, tenths, hundredths and thousandths) as decimals and
percentages. For example, 3/8 = 0.375 = 37.5% and use whatever form is easiest for a given calculation, e.g. 30% of $78 as 3/10 of 78 = . Students should also be able to give the fraction form of any decimals to three places and vice versa,
e.g. 1.346 = 1346/1000, and express percentages, including those greater than one hundred, as decimals and vice versa, e.g. 1.75 as 175%.
AO6: Know and apply standard form, significant figures, rounding, and decimal place value.
This means students will be able to express a given whole number or decimal measurements in standard form and vice versa and understand the potential rounding that may be involved. Standard form (scientific notation) at this level should involve integral exponents, e.g. 24 300 = 2.43 x 104 or 0.0243 = 2.43 x 10-2. This understanding of decimal place value and rounding should include interpretation of the potential value of a measurement when it is expressed using significant figures, e.g. 2.3m (2sf.) has a potential measurement of 2. 25≤m<2.35 whereas 2.30 (4sf.) has a potential measurement of 2.295≤m<2.305. Students should also apply decimal place value and sensible rounding through estimating in a way that is suitable to the context, and recognising the effects of that rounding on the accuracy of the estimation, e.g. 48.7 ÷ 2.13 = can be estimated by 50 ÷ 2 = 25 but the rounding results in an estimate that is too high. This does not include putting error bounds on estimations.
Equations and expressions
AO1: Form and solve linear and simple quadratic equations.
Students should be able to form the linear equation or simple quadratic (y = ax2 or y = x2 ± c, a and c are integers) to model a given situation (see patterns and relationships). They should understand that solving an equation involves finding the value of a variable when the other variable is defined, and interpret how the solution relates to the original context. Students should be able to solve linear and simple quadratic equations by applying inverse operations with an understanding of the equals sign as a statement of transitive balance, for example (3q + 7)/4 = 16, by multiplying both sides by four, subtracting seven, etc. They should also recognise where it is appropriate to solve an equation through trial and improvement, and find the missing value by systematic calculation.
Patterns and relationships
AO1: Generalise the properties of operations with fractional numbers and integers.
This means students will understand that to generalise 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. Fractional numbers, for the purpose of this objective, are defined as rational numbers in the form a/b,
where a and b are whole numbers and b ? 0. At Level Five students should be able to demonstrate their understanding of the properties of addition, subtraction,
multiplication and division as these operations apply to fractional numbers and integers. These properties include commutativity, distributivity, associativity,
inverse and identity. Demonstration of understanding should involve applying these properties in solving a variety of problems, using the properties to solve
equations without calculating both sides, for example, 6 x = 3 x 70 + 3 x 9, justifying their responses to conjectures such as true/false statements, and expressing the generalisations algebraically, for example, the commutative property for multiplication of integers may be represented by a x b = b x a, where a and b are integers. Students should be able to express the operations on fractional numbers algebraically, for example, a/b + c/d = (ad + cb)/bd, and substitute number values into the equation to confirm that it holds for all addition examples they attempt.
AO2: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.
This means students will recognise the features of tabular, graph and equation representations of linear and quadratic relationships. This includes connecting constant first or second order difference in tables with linear and quadratic relations respectively, with the graph (linear and parabolic) and standard equation forms (y = mx + c and y = a x 2 + bx + c) for such relations. For example, given the spatial pattern below students can use a table, graph or equation to represent the relation and solve problems.
This includes finding both a recursive and direct (functional) rules and using them to find further terms using a spreadsheet or calculator, for example:
Students should also use these tabular and graphic representations for other relationships, such as simple exponential and step relations, but it is acceptable for them to use recursive rules for these more difficult relations.
Click to download a PDF of second-tier material relating to Level 5 Patterns and Relationships (527KB)
Geometry and Measurement
Measurement
AO1: Select and use appropriate metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time, with awareness that measurements are approximate. Further detail on this Achievement Objective will be added shortly.
AO2: Convert between metric units, using decimals. Further detail on this Achievement Objective will be added shortly.
AO3: Deduce and use formulae to find the perimeters and areas of polygons and the volumes of prisms. Further detail on this Achievement Objective will be added shortly.
AO4: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders. Further detail on this Achievement Objective will be added shortly.
Click to download a PDF of second-tier material relating to Level 5 Measurement (207KB)
Shape
AO1: Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties. Further detail on this Achievement Objective will be added shortly.
AO2: Create accurate nets for simple polyhedra and connect three-dimensional solids with different two-dimensional representations. Further detail on this Achievement Objective will be added shortly.
Position and orientation
AO1: Construct and describe simple loci. Further detail on this Achievement Objective will be added shortly.
AO2: Interpret points and lines on co-ordinate planes, including scales and bearings on maps. Further detail on this Achievement Objective will be added shortly.
Transformation
AO1: Define and use transformations and describe the invariant properties of figures and objects under these transformations. Further detail on this Achievement Objective will be added shortly.
AO2: Apply trigonometric ratios and Pythagoras’ theorem in two dimensions. Further detail on this Achievement Objective will be added shortly.
Statistics
Statistical investigation
AO1: Plan and conduct surveys and experiments using the statistical enquiry
cycle:
Further detail on this Achievement Objective will be added shortly.
Click to download a PDF of second-tier material relating to Level 5 Statistical Investigations (288KB)
Statistical literacy
AO1: Evaluate statistical investigations or probability activities undertaken by others, including data collection methods, choice of measures, and validity of findings. Further detail on this Achievement Objective will be added shortly.
Probability
AO1: Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance. Further detail on this Achievement Objective will be added shortly.
AO2: Calculate probabilities, using fractions, percentages, and ratios. Further detail on this Achievement Objective will be added shortly.
