NSW Syllabuses

# Mathematics K–10 - Stage 1 - Number and Algebra Multiplication and Division

## Multiplication and Division 1

### Outcomes

#### A student:

• MA1-1WM

describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols

• MA1-6NA

uses a range of mental strategies and concrete materials for multiplication and division

### Content

• Students:
• Skip count by twos, fives and tens starting from zero (ACMNA012)
• count by twos, fives and tens using rhythmic counting and skip counting from zero
• use patterns on a number chart to assist in counting by twos, fives or tens (Communicating)
• Model and use equal groups of objects as a strategy for multiplication
• model and describe collections of objects as 'groups of', eg
• recognise the importance of having groups of equal size (Reasoning)
• determine and distinguish between the 'number of groups' and the 'number in each group' when describing collections of objects (Communicating)
• find the total number of objects using skip counting
• Recognise and represent division as grouping into equal sets (ACMNA032)
• recognise when there are equal numbers of items in groups, eg 'There are three pencils in each group'
• model division by sharing a collection of objects equally into a given number of groups to determine how many in each group, eg determine the number in each group when 10 objects are shared between two people
• describe the part left over when a collection cannot be shared equally into a given number of groups (Communicating, Problem Solving, Reasoning)
• model division by sharing a collection of objects into groups of a given size to determine the number of groups, eg determine the number of groups when 20 objects are shared into groups of four
• describe the part left over when a collection cannot be distributed equally using the given group size, eg when 22 objects are shared into groups of four, there are five groups of four and two objects left over (Communicating, Problem Solving, Reasoning)

### Background Information

There are two forms of division:

Sharing (partitive) – How many in each group?
eg 'If 12 marbles are shared between three students, how many does each get?'

Grouping (quotitive) – How many groups are there?
eg 'If I have 12 marbles and each child is to get four, how many children will get marbles?'

After students have divided a quantity into equal groups (eg they have divided 12 into groups of four), the process can be reversed by combining the groups, thus linking multiplication and division.

When sharing a collection of objects into two groups, students may describe the number of objects in each group as being one-half of the whole collection.

### Language

Students should be able to communicate using the following language: group, number of groups, number in each group, sharing, shared between, left over, total, equal.

Sharing – relates to distributing items one at a time into a set number of groups, eg the student has a number of pop sticks and three cups and shares out the pop sticks into the cups one at a time.

Grouping – relates to distributing the same number of items into an unknown number of groups, eg the student has 12 pop sticks and wants to make groups of four, so places four pop sticks down, then another four, and so on.

It is preferable that students use 'groups of', before progressing to 'rows of' and 'columns of'. The term 'lots of' can be confusing to students because of its everyday use and should be avoided, eg 'lots of fish in the sea'.

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

When working towards the outcome MA1‑6NA the sub-elements (and levels) of Quantifying numbers (QuN7), Additive strategies (AdS6), Multiplicative strategies (MuS2-MuS4), Number patterns and algebraic thinking (NPA4) and Interpreting fractions (InF1) 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.

## Multiplication and Division 2

### Outcomes

#### A student:

• MA1-1WM

describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols

• MA1-2WM

uses objects, diagrams and technology to explore mathematical problems

• MA1-3WM

supports conclusions by explaining or demonstrating how answers were obtained

• MA1-6NA

uses a range of mental strategies and concrete materials for multiplication and division

### Content

• Students:
• Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)
• model multiplication as repeated addition, eg 3 groups of 4 is the same as 4 + 4 + 4
• find the total number of objects by placing them into equal-sized groups and using repeated addition (Problem Solving)
• use empty number lines and number charts to record repeated addition, eg
(Communicating)
• explore the use of repeated addition to count in practical situations, eg counting stock on a farm (Problem Solving)
• recognise when items have been arranged into groups, eg 'I can see two groups of three pencils'
• use concrete materials to model multiplication as equal 'groups' and by forming an array of equal 'rows' or equal 'columns', eg
• describe collections of objects as 'groups of', 'rows of' and 'columns of' (Communicating)
• determine and distinguish between the 'number of rows/columns' and the 'number in each row/column' when describing collections of objects (Communicating)
• recognise practical examples of arrays, such as seedling trays or vegetable gardens (Reasoning)
• model the commutative property of multiplication, eg '3 groups of 2 is the same as 2 groups of 3'
• Represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)
• model division by sharing a collection of objects equally into a given number of groups, and by sharing equally into a given number of rows or columns in an array, eg determine the number each person receives when 10 objects are shared between two people
• describe the part left over when a collection cannot be shared equally into a given number of groups/rows/columns (Communicating, Problem Solving, Reasoning)
• model division by sharing a collection of objects into groups of a given size, and by arranging it into rows or columns of a given size in an array, eg determine the number of columns in an array when 20 objects are arranged into rows of four
• describe the part left over when a collection cannot be distributed equally using the given group/row/column size, eg when 14 objects are arranged into rows of five, there are two rows of five and four objects left over (Communicating, Problem Solving, Reasoning)
• model division as repeated subtraction
• use an empty number line to record repeated subtraction (Communicating)
• explore the use of repeated subtraction to share in practical situations, eg share 20 stickers between five people (Problem Solving)
• solve multiplication and division problems using objects, diagrams, imagery and actions
• recognise which strategy worked and which did not work and explain why (Communicating, Reasoning)
• record answers to multiplication and division problems using drawings, words and numerals, eg 'two rows of five make ten', '2 rows of 5 is 10'

### Background Information

There are two forms of division:

Sharing (partitive) – How many in each group?
eg 'If 12 marbles are shared between three students, how many does each get?'

Grouping (quotitive) – How many groups are there?
eg 'If I have 12 marbles and each child is to get four, how many children will get marbles?' This form of division relates to repeated subtraction, 12 – 4 – 4 – 4 = 0, so three children will get four marbles each.

After students have divided a quantity into equal groups (eg they have divided 12 into groups of four), the process can be reversed by combining the groups, thus linking multiplication and division.

When sharing a collection of objects into two, four or eight groups, students may describe the number of objects in each group as being one-half, one-quarter or one-eighth, respectively, of the whole collection.

An array is one of several different arrangements that can be used to model multiplicative situations involving whole numbers. It is made by arranging a set of objects, such as counters, into columns and rows. Each column must contain the same number of objects as the other columns, and each row must contain the same number of objects as the other rows.

Formal writing of number sentences for multiplication and division, including the use of the symbols × and ÷, is not introduced until Stage 2.

### Language

Students should be able to communicate using the following language: add, take away, group, row, column, array, number of rows, number of columns, number in each row, number in each column, total, equal, is the same as, shared between, shared equally, part left over, empty number line, number chart.

The term 'row' refers to a horizontal grouping, and the term 'column' refers to a vertical grouping.

Refer also to language in Stage 1 Multiplication and Division 1.

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

When working towards the outcome MA1‑6NA the sub-elements (and levels) of Additive strategies (AdS6), Multiplicative strategies (MuS4-MuS5), Number patterns and algebraic thinking (NPA5) and Interpreting fractions (InF1) 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.