Patterns and Algebra 1
Outcomes
A student:

 MA31WM
describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

 MA32WM
selects and applies appropriate problemsolving strategies, including the use of digital technologies, in undertaking investigations

 MA33WM
gives a valid reason for supporting one possible solution over another

 MA38NA
analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane
 Students:
 Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)
 identify, continue and create simple number patterns involving addition and subtraction
 describe patterns using the terms 'increase' and 'decrease', eg for the pattern 48, 41, 34, 27, …, 'The terms decrease by seven'
 create, with materials or digital technologies, a variety of patterns using whole numbers, fractions or decimals, eg \( \frac{1}{4}, ~\frac{2}{4}, ~\frac{3}{4}, ~\frac{4}{4}, ~\frac{5}{4}, ~\frac{6}{4}, \) … or 2.2, 2.0, 1.8, 1.6, …
 use a number line or other diagram to create patterns involving fractions or decimals
 Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)
 complete number sentences that involve more than one operation by calculating missing numbers, eg \( 5 \times \square = 4 \times 10 \), \( 5 \times \square = 30  10 \)
 describe strategies for completing simple number sentences and justify solutions (Communicating, Reasoning)
 identify and use inverse operations to assist with the solution of number sentences, eg \( 125 \div 5 = \square \) becomes \( \square \times 5 = 125 \)
 describe how inverse operations can be used to solve a number sentence (Communicating, Reasoning)
 complete number sentences involving multiplication and division, including those involving simple fractions or decimals, eg \( 7 \times \square = 7.7 \)
 check solutions to number sentences by substituting the solution into the original question (Reasoning)
 write number sentences to match word problems that require finding a missing number, eg 'I am thinking of a number that when I double it and add 5, the answer is 13. What is the number?'
Background Information
Students should be given opportunities to discover and create patterns and to describe, in their own words, relationships contained in those patterns.
This substrand involves algebra without using letters to represent unknown values. When calculating unknown values, students need to be encouraged to work backwards and to describe the processes using inverse operations, rather than using trialanderror methods. The inclusion of number sentences that do not have wholenumber solutions will aid this process.
To represent equality of mathematical expressions, the terms 'is the same as' and 'is equal to' should be used. Use of the word 'equals' may suggest that the righthand side of an equation contains 'the answer', rather than a value equivalent to that on the left.
Language
Students should be able to communicate using the following language: pattern, increase, decrease, missing number, number sentence, number line.
In Stage 3, students should be encouraged to use their own words to describe number patterns. Patterns can usually be described in more than one way and it is important for students to hear how other students describe the same pattern. Students' descriptions of number patterns can then become more sophisticated as they experience a variety of ways of describing the same pattern. The teacher could begin to model the use of more appropriate mathematical language to encourage this development.
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA3‑8NA the subelements (and levels) of Number patterns and algebraic thinking (NPA4NPA6) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.
Patterns and Algebra 2
Outcomes
A student:

 MA31WM
describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

 MA32WM
selects and applies appropriate problemsolving strategies, including the use of digital technologies, in undertaking investigations

 MA33WM
gives a valid reason for supporting one possible solution over another

 MA38NA
analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane
 Students:
 Continue and create sequences involving whole numbers, fractions and decimals; describe the rule used to create the sequence (ACMNA133)
 continue and create number patterns, with and without the use of digital technologies, using whole numbers, fractions and decimals, eg \( \frac{1}{4},~ \frac{1}{8},~ \frac{1}{16},~ \ldots \) or 1.25, 2.5, 5, …
 describe how number patterns have been created and how they can be continued (Communicating, Problem Solving)

create simple geometric patterns using concrete materials, eg
\( \triangle \:,\:\triangle\triangle \:,\:\triangle\triangle\triangle \:,\:\triangle\triangle\triangle\triangle \:,\: ~\ldots \) 
complete a table of values for a geometric pattern and describe the pattern in words, eg
 describe the number pattern in a variety of ways and record descriptions using words, eg 'It looks like the multiplication facts for four'
 determine the rule to describe the pattern by relating the bottom number to the top number in a table, eg 'You multiply the number of squares by four to get the number of matches'
 use the rule to calculate the corresponding value for a larger number, eg 'How many matches are needed to create 100 squares?'

complete a table of values for number patterns involving one operation (including patterns that decrease) and describe the pattern in words, eg
 describe the pattern in a variety of ways and record descriptions in words, eg 'It goes up by ones, starting from four'
 determine a rule to describe the pattern from the table, eg 'To get the value of the term, you add three to the position in the pattern'
 use the rule to calculate the value of the term for a large position number, eg 'What is the 55th term of the pattern?'
 explain why it is useful to describe the rule for a pattern by describing the connection between the 'position in the pattern' and the 'value of the term' (Communicating, Reasoning)
 interpret explanations written by peers and teachers that accurately describe geometric and number patterns (Communicating)
 make generalisations about numbers and number relationships, eg 'If you add a number and then subtract the same number, the result is the number you started with'
 Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)
 recognise that the number plane (Cartesian plane) is a visual way of describing location on a grid

recognise that the number plane consists of a horizontal axis (\(x\)axis) and a vertical axis (\(y\)axis), creating four quadrants
 recognise that the horizontal axis and the vertical axis meet at right angles (Reasoning)
 identify the point of intersection of the two axes as the origin, having coordinates (0, 0)
 plot and label points, given coordinates, in all four quadrants of the number plane
 plot a sequence of coordinates to create a picture (Communicating)
 identify and record the coordinates of given points in all four quadrants of the number plane
 recognise that the order of coordinates is important when locating points on the number plane, eg (2, 3) is a location different from (3, 2) (Communicating)
Background Information
Refer to background information in Patterns and Algebra 1.
In Stage 2, students found the value of the next term in a pattern by performing an operation on the previous term. In Stage 3, they need to connect the value of a particular term in the pattern with its position in the pattern. This is best achieved through a table of values. Students need to see a connection between the two numbers in each column and should describe the pattern in terms of the operation that is performed on the position in the pattern to obtain the value of the term. Describing a pattern by the operation(s) performed on the 'position in the pattern' is more powerful than describing it as an operation performed on the previous term in the pattern, as it allows any term (eg the 100th term) to be calculated without needing to find the value of the term before it. The concept of relating the number in the top row of a table of values to the number in the bottom row forms the basis for work in Linear and NonLinear Relationships in Stage 4 and Stage 5.
The notion of locating position and plotting coordinates is established in the Position substrand in Stage 2 Measurement and Geometry. It is further developed in this substrand to include negative numbers and the use of the fourquadrant number plane.
The Cartesian plane (commonly referred to as the 'number plane') is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first to develop analytical geometry on the number plane. On the number plane, the 'coordinates of a point' refers to the ordered pair \( (x,y) \) describing the horizontal position \(x\) first, followed by the vertical position \(y\).
The Cartesian plane is applied in realworld contexts, eg when determining the incline (slope) of a road between two points.
The Cartesian plane is used in algebra in Stages 4 to 6 to describe patterns and relationships between numbers.
Language
Students should be able to communicate using the following language: pattern, increase, decrease, term, value, table of values, rule, position in pattern, value of term, number plane (Cartesian plane), horizontal axis (xaxis), vertical axis (yaxis), axes, quadrant, intersect, point of intersection, right angles, origin, coordinates, point, plot.
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA3‑8NA the subelements (and levels) of Number patterns and algebraic thinking (NPA4NPA7) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.