NSW Syllabuses

# Mathematics K–10 - Stage 3 - Number and Algebra Whole Numbers

## Whole Numbers 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

• MA3-4NA

orders, reads and represents integers of any size and describes properties of whole numbers

### Content

• Students:
• Recognise, represent and order numbers to at least tens of millions
• apply an understanding of place value and the role of zero to read and write numbers of any size
• state the place value of digits in numbers of any size
• arrange numbers of any size in ascending and descending order
• record numbers of any size using expanded notation, eg 163 480 = 100 000 + 60 000 + 3000 + 400 + 80
• partition numbers of any size in non-standard forms to aid mental calculation, eg when adding 163 480 and 150 000, 163 480 could be partitioned as 150 000 + 13 480, so that 150 000 could then be doubled and added to 13 480
• use numbers of any size in real-life situations, including in money problems
• interpret information from the internet, the media, the environment and other sources that use large numbers (Communicating, Reasoning)
• recognise different abbreviations of numbers used in everyday contexts, eg $350 K represents$350 000
• round numbers to a specified place value, eg round 5 461 883 to the nearest million
• determine all 'factors' of a given whole number, eg 36 has factors 1, 2, 3, 4, 6, 9, 12, 18 and 36
• determine the 'highest common factor' (HCF) of two whole numbers, eg the HCF of 16 and 24 is 8
• determine 'multiples' of a given whole number, eg multiples of 7 are 7, 14, 21, 28, …
• determine the 'lowest common multiple' (LCM) of two whole numbers, eg the LCM of 21 and 63 is 63
• determine whether a particular number is a factor of a given number using digital technologies
• recognise that when a given number is divided by one of its factors, the result must be a whole number (Problem Solving)
• solve problems using knowledge of factors and multiples, eg 'There are 48 people at a party. In how many ways can you set up the tables and chairs, so that each table seats the same number of people and there are no empty chairs?'

### Background Information

Students need to develop an understanding of place value relationships, such as 10 thousand = 100 hundreds = 1000 tens = 10 000 ones.

### Language

Students should be able to communicate using the following language: ascending order, descending order, zero, ones, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, digit, place value, expanded notation, round to, whole number, factor, highest common factor (HCF), multiple, lowest common multiple (LCM).

In some Asian languages, such as Chinese, Japanese and Korean, the natural language structures used when expressing numbers larger than 10 000 are 'tens of thousands' rather than 'thousands', and 'tens of millions' rather than 'millions'. For example, in Chinese (Mandarin), 612 000 is expressed as '61 wàn, 2 qiān', which translates as '61 tens of thousands and 2 thousands'.

The abbreviation 'K' is derived from the Greek word khilios, meaning 'thousand'. It is used in many job advertisements to represent salaries (eg a salary of $70 K or$70 000). It is also used as an abbreviation for the size of computer files (eg a size of 20 K, meaning twenty thousand bytes).

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑4NA the sub-elements (and levels) of Quantifying numbers (QuN11-QuN12), Additive strategies (AdS7-AdS8), Multiplicative strategies (MuS5-MuS6) and Understanding money (UnM7) 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.

## Whole Numbers 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

• MA3-4NA

orders, reads and represents integers of any size and describes properties of whole numbers

### Content

• Students:
• Investigate everyday situations that use integers; locate and represent these numbers on a number line (ACMNA124)
• recognise the location of negative whole numbers in relation to zero and place them on a number line
• use the term 'integers' to describe positive and negative whole numbers and zero
• interpret integers in everyday contexts, eg temperature
• investigate negative whole numbers and the number patterns created when counting backwards on a calculator
• recognise that negative whole numbers can result from subtraction (Reasoning)
• ask 'What if' questions, eg 'What happens if we subtract a larger number from a smaller number on a calculator?' (Communicating)
• determine whether a number is prime, composite or neither
• explain whether a whole number is prime, composite or neither by finding the number of factors, eg '13 has two factors (1 and 13) and therefore is prime', '21 has more than two factors (1, 3, 7, 21) and therefore is composite', '1 is neither prime nor composite as it has only one factor, itself' (Communicating, Reasoning)
• explain why a prime number, when modelled as an array, can have only one row (Communicating, Reasoning)
• model square and triangular numbers and record each number group in numerical and diagrammatic form
• explain how square and triangular numbers are created (Communicating, Reasoning)
• explore square and triangular numbers using arrays, grid paper or digital technologies (Communicating, Problem Solving)
• recognise and explain the relationship between the way each pattern of numbers is created and the name of the number group (Communicating, Reasoning)

### Background Information

Students could investigate further the properties of square and triangular numbers, such as all square numbers have an odd number of factors, while all non-square numbers have an even number of factors; when two consecutive triangular numbers are added together, the result is always a square number.

### Language

Students should be able to communicate using the following language: number line, whole number, zero, positive number, negative number, integer, prime number, composite number, factor, square number, triangular number.

Words such as 'square' have more than one grammatical use in mathematics, eg draw a square (noun), square three (verb), square numbers (adjective) and square metres (adjective).

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑4NA the sub-elements (and levels) of Quantifying numbers (QuN12) 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.