Number and Algebra: Patterns and relationships, Level 6
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.
