Number and Algebra: Patterns and relationships, Level 3
AO1: Generalise the properties of addition and subtraction with whole numbers.
This means students will generalise, which means to establish properties that hold for all instances.
Generalisation begins with noticing patterns and relationships in a few specific instances, defining the
variables involved, noticing the relationships between the variables, then using appropriate mathematical
terminology and symbols to describe the relationships. At Level Three students develop many generalisations
that allow them to perform mental strategies effectively. These generalisations include, the commutative property of addition and multiplication, e.g. 7 x 8 = 8 x 7, the associative property of addition and multiplication, e.g. (2 x 3) x 4 = 2 x (3 x 4), the distributive property of multiplication, e.g. 8 x 7 = 8 x 5 + 8 x 2, the inverse relationships of addition and subtraction, and of multiplication and division, e.g. 6 x 7 = 42 so 42 ÷ 7 = 6, and identities for all four operations, e.g. 17 x 1 = 17, 17 ÷ 1 = 17. It is not expected that students use algebraic symbols to express these generalisations. However, students should be able to look for
relationships across the equals sign in equations to determine missing numbers, e.g. 4 x 12 =
x 6 without calculating 4 x 12.
AO2: Connect members of sequential patterns with their ordinal position and use
tables, graphs, and diagrams to find relationships between successive
elements of number and spatial patterns.
This means students will recognise that a sequential pattern can be either spatial,
e.g. 
..,
or numeric, e.g. 1, 3, 5, 7, ... A pattern has consistency so further terms of it can be anticipated from those already known.
The focus in this thread is that students become increasingly sophisticated at describing the relationships between variables found
in sequential patterns. With spatial patterns, students at Level Three should be able to identify the repeating element, e.g.
, and use simple multiplicative thinking
to predict the shape in a given ordinal position, e.g.
Every third shape is 
so the thirtieth shape will be
so the thirty-second shape will be 
With number patterns students should identify the consistent relationship between variables in simple multiple situations,
e.g. 4, 8, 12, 16,... are all multiples of four, or identify the additive “gap” between the terms,
e.g. 4, 7, 10, 13,... three is added each time.
They should be able to describe these rules in their own words and use their rules to find further terms.
Students also use tables, graphs, diagrams and word rules to find and describe relationships in patterns, e.g.
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“There is always one more peg that the number of towels. The first towel took two pegs.”
Click to download a PDF of second-tier material relating to Level 3 Patterns and Relationships (319KB)
