NSW Syllabuses

# Mathematics K–10 - Stage 3 - Number and Algebra Patterns and Algebra

## Patterns and Algebra 1

### Outcomes

#### A student:

• MA3-1WM

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

• MA3-2WM

selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations

• MA3-3WM

gives a valid reason for supporting one possible solution over another

• MA3-8NA

analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane

### Content

• 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 trial-and-error methods. The inclusion of number sentences that do not have whole-number 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 right-hand 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 sub-elements (and levels) of Number patterns and algebraic thinking (NPA4-NPA6) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.

## Patterns and Algebra 2

### Outcomes

#### A student:

• MA3-1WM

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

• MA3-2WM

selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations

• MA3-3WM

gives a valid reason for supporting one possible solution over another

• MA3-8NA

analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane

### Content

• 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'
• 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 Non-Linear 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 four-quadrant 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 real-world 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 (x-axis), vertical axis (y-axis), 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 sub-elements (and levels) of Number patterns and algebraic thinking (NPA4-NPA7) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.