Number strategies and knowledge

NA5-1: 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, that is, 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 includes many of the following contexts:

• 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.
  Supporting teaching resources.

NA5-2: 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, for example 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 a^b x a^c = a^b+c and a^b ÷ a^c = a^{b - c} and compare powers relationally (without calculation) where this is appropriate, for example 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. Supporting teaching resources.

NA5-3: Understand operations on fractions, decimals, percentages, and integers. 

This means students will understand calculations involving fractions, decimals, percentages and integers. It 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 implies 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. Supporting teaching resources.

NA5-4: 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 (for example 4 x ☐ = 25) or between (for example 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 weaker 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. Supporting teaching resources.

NA5-5: 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, for example 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, for example 1.346 = 1346/1000, and express percentages, including those greater than one hundred, as decimals and vice versa, for example 1.75 as 175%. Supporting teaching resources.

NA5-6: 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, for example 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, for example 2.3m (2sf.) has a potential measurement of 2. 25≤m<2.35 whereas 2.300 (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, for example 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. Supporting teaching resources.

Equations and expressions

NA5-7: 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. Supporting teaching resources.

Patterns and relationships

NA5-8: 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. Supporting teaching resources.

NA5-9: 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. Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 5 Patterns and Relationships (527KB)