Number and Algebra: Patterns and relationships

Level 1

AO1: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.
This means students will understand the link between the cardinal and ordinal aspects of counting. The ordinal aspect refers to the fact that counting numbers have a conventional order. The last number in a count tells how many objects are in a set if all the objects are matched in one-to-one correspondence to the sequence of counting numbers. s. The next number in the counting sequence tells the result of adding an object while the number before in the sequence tells the count when an object is removed. The cardinal aspect involves knowing that when counting a set of items the last number describes all the items in the set, no matter their colour, size, arrangement or other attributes. This count can be trusted and built upon.

AO2: Create and continue sequential patterns.
This means the students will explore sequential patterns. A sequential pattern is one in which further members of that pattern can be predicted from previous members. So ..., and 1, 3, 5, 7, ... are sequential patterns. At Level One students should be able to reproduce a given pattern using objects, drawings or symbols and continue the pattern on with justification, e.g. It goes square, circle, star...They should also be able to invent their own patterns and communicate the “rule” for their pattern to others.

Click to download a PDF of second-tier material relating to Level 1 Patterns and Relationships (288KB)

Level 2

AO1: Generalise that whole numbers can be partitioned in many ways.

Students at level two should understand that numbers are counts that can be split in ways that make the operations of addition, subtraction, multiplication and division easier. From Level One students understand that counting a set tells how many objects are in the set. An advance on this thinking is to realise that the count of a set can be partitioned and that the count of each subset tells how many objects are in that subset. Also required is understanding that partitions of a count can be recombined. For example, a count of ten can be partitioned into 1 and 9, 2 and 8, 3 and 7, etc. This objective also involves critical choice of partitioning. For example, 8 + 6 =

can be solved by partitioning 6 into 1 and 5, 2 and 4, 3 and 3. Of these partitions 2 and 4 is the best strategic choice since it recombines into a “ten and...” fact, i.e. 8 + 6 = 8 + 2 + 4 = 10 + 4. At Level Two students are expected to understand the strategic importance of using place value as a way to partition numbers. Students should apply their partitioning generalisation to many problem types including combining (e.g. 27 + 9 =

) , separating (e.g. 105 - 19 =

), comparing (e.g. 45 +

= 106), duplicating (e.g. 8 x 5 =

) and sharing (e.g. 20 ÷ 4 =

AO2: Find rules for the next member in a sequential pattern.
This means students will explore sequential patterns, both can be either spatial, e.g. ,.. or numeric, e.g. 1, 3, 5, 7, ... A pattern has consistency so further ter ms of it can be anticipated from those already known. In spatial patterns students should be able to identify the repeating element, e.g. in that above, and use this to predict the shape in a given ordinal position, e.g. the next shape is , the eleventh shape will be . For simple number patterns students should identify the consistent “gap” between the terms, e.g. 1, 3, 5, 7,... two is added each time, and use this additive difference to find further terms. Students should also develop their concept of relations between variables using spatial patterns that can be represented using numeric tables of values, e.g. For this pattern, how many squares make 7 crosses?

1 cross
5 squares 2 crosses
10 squares 3 crosses
15 squares

Click to download a PDF of second-tier material relating to Level 2 Patterns and Relationships (203KB)

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.

Towels	Pegs
1	2
2	3
3	4
4	5

pegsgraph.

“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)

Level 4

AO1: Generalise properties of multiplication and division with whole numbers.
This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to look at several examples, notice what changes (variables) and what does not, use appropriate mathematical terminology and symbols to describe the pattern, and apply the generalisation to other examples. At Level Four students should be able to describe and apply the properties of multiplication and division as these operations apply to whole numbers. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express generalisations using words and symbols, e.g. 4 x 6 = 24 so 24 ÷ 6 = 4 and 24 ÷ 4 = 6 (example) leading to a x b = c so c ÷ b = a and c ÷ a = b. This is the inverse relationship of multiplication and division.

AO2: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
This means students will describe the function rule for a linear relationship as well as recognise recursive relationships where more complex relationships are involved. For example, given the pattern of fish made with matchsticks and counters below, students should be able to represent the relationships in a table and graph and use these representations to predict the terms in the sequence:

Counters	1	2	3	4	5
Matchsticks	8	14	20	26	32

Level Four students should be able to:

Give linear rules connecting the variables, e.g. "the number of matchsticks is the six times the number of counters plus two", or "take one off the number of fish, multiply that number by six then add eight".
Extend the graph or table of a linear relationship to predict further co-ordinate pairs, recognising that constant difference (add six in the fish pattern) is associated with points that lay on a line.
Use recursive methods to predict further members of a sequence where the relationship is non-linear, e.g. The sequence of triangular numbers:

Number of rows	1	2	3	4	5	...	10
Total of counters	1	3	6	10	15	...	55

Recursive means finding what is added to or subtracted from one term to get the next.

Click to download a PDF of second-tier material relating to Level 4 Patterns and Relationships (282KB)

Level 5

AO1: 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.

AO2: 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.

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

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)/ 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.

TKI main navigation