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Number and Algebra: Number strategies and knowledge, Level 6

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, 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:

  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
table.