Elaborations on Level Six: 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:

1. 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.
2. 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).
3. 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:

1. 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.
2. 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.
3. 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, a^b x a = a^{b + c} and a^b ÷ a^c = a^{b - 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 a^b = a^{b + -b}. Since a^{b + -b} = a⁰ = 1, a^-b must be the reciprocal of a^b ( 1/a^b) since a^b x 1/a^b = 1. For example, 6³ = 216 so 6^-3 = 1/6³ = 1/216. The meaning of a^1/b must preserve the truth of (a^1/b)^b = 1, e.g. a^1/2x a^1/2 = a¹ = a, so a^1/2 must be the square root of a (√a) and a^1/3x a^1/3 x a^1/3 = a¹1 = a, so a^1/3 must be the cube root of a (³√a). So in general a^1/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 4^{3 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:
 

1. 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.
2. Create formulae that calculate the quantities required by the problem, e.g. Length of Edge C and surface area in this problem.
3. 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. 3^x = 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)/ a^m = a^{n - 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:

Year One	Year Two	Year Three

The relationship between years and matches could be expressed as:

Year Matches Difference 1 4   2 12 8 3 24 12 4 40 16 5 60 20 6 84 24 7 112 28 8 144 32 9 180 36 10 220 40

...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.