NSW Syllabuses

# Mathematics K–10 - Stage 2 - Number and Algebra Multiplication and Division

## Multiplication and Division 1

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-2WM

selects and uses appropriate mental or written strategies, or technology, to solve problems

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-6NA

uses mental and informal written strategies for multiplication and division

### Content

• Students:
• Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)
• count by twos, threes, fives or tens using skip counting
• use mental strategies to recall multiplication facts for multiples of two, three, five and ten
• relate 'doubling' to multiplication facts for multiples of two, eg 'Double three is six' (Reasoning)
• recognise and use the symbols for multiplied by (×), divided by (÷) and equals (=)
• link multiplication and division facts using groups or arrays, eg
• explain why a rectangular array can be read as a division in two ways by forming vertical or horizontal groups, eg 12 ÷ 3 = 4 or 12 ÷ 4 = 3 (Communicating, Reasoning)
• model and apply the commutative property of multiplication, eg 5 × 8 = 8 × 5
• Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)
• use mental strategies to multiply a one-digit number by a multiple of 10, including:
• repeated addition, eg 3 × 20: 20 + 20 + 20 = 60
• using place value concepts, eg 3 × 20: 3 × 2 tens = 6 tens = 60
• factorising the multiple of 10, eg 3 × 20: 3 × 2 × 10 = 6 × 10 = 60
• apply the inverse relationship of multiplication and division to justify answers, eg 12 ÷ 3 is 4 because 4 × 3 = 12 (Reasoning)
• select, use and record a variety of mental strategies, and appropriate digital technologies, to solve simple multiplication problems
• pose multiplication problems and apply appropriate strategies to solve them (Communicating, Problem Solving)
• explain how an answer was obtained and compare their own method of solution with the methods of other students (Communicating, Reasoning)
• explain problem-solving strategies using language, actions, materials and drawings (Communicating, Problem Solving)
• describe methods used in solving multiplication problems (Communicating)

### Background Information

In Stage 2, the emphasis in multiplication and division is on students developing mental strategies and using their own (informal) methods for recording their strategies. Comparing their own method of solution with the methods of other students will lead to the identification of efficient mental and written strategies. One problem may have several acceptable methods of solution.

Students could extend their recall of number facts beyond the multiplication facts to 10 × 10 by also memorising multiples of numbers such as 11, 12, 15, 20 and 25.

An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations.

The use of digital technologies includes the use of calculators.

### Language

Students should be able to communicate using the following language: group, row, column, horizontal, vertical, array, multiply, multiplied by, multiplication, multiplication facts, double, shared between, divide, divided by, division, equals, strategy, digit, number chart.

When beginning to build and read multiplication facts aloud, it is best to use a language pattern of words that relates back to concrete materials such as arrays. As students become more confident with recalling multiplication facts, they may use less language. For example, 'five rows (or groups) of three' becomes 'five threes' with the 'rows of' or 'groups of' implied. This then leads to 'one three is three', 'two threes are six', 'three threes are nine', and so on.

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

When working towards the outcome MA2‑6NA the sub-elements (and levels) of Multiplicative strategies (MuS2-MuS6) and Number patterns and algebraic thinking (NPA5-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.

## Multiplication and Division 2

### Outcomes

#### A student:

• MA2-1WM

uses appropriate terminology to describe, and symbols to represent, mathematical ideas

• MA2-2WM

selects and uses appropriate mental or written strategies, or technology, to solve problems

• MA2-3WM

checks the accuracy of a statement and explains the reasoning used

• MA2-6NA

uses mental and informal written strategies for multiplication and division

### Content

• Students:
• Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)
• count by fours, sixes, sevens, eights and nines using skip counting
• use the term 'product' to describe the result of multiplying two or more numbers, eg 'The product of 5 and 6 is 30'
• use mental strategies to build multiplication facts to at least 10 × 10, including:
• using the commutative property of multiplication, eg 7 × 9 = 9 × 7
• using known facts to work out unknown facts, eg 5 × 7 is 35, so 6 × 7 is 7 more, which is 42
• using doubling and repeated doubling as a strategy to multiply by 2, 4 and 8, eg 7 × 8 is double 7, double again and then double again
• using the relationship between multiplication facts, eg the multiplication facts for 6 are double the multiplication facts for 3
• factorising one number, eg 5 × 8 is the same as 5 × 2 × 4, which becomes 10 × 4
• recall multiplication facts up to 10 × 10, including zero facts, with automaticity
• find 'multiples' for a given whole number, eg the multiples of 4 are 4, 8, 12, 16, …
• relate multiplication facts to their inverse division facts, eg 6 × 4 = 24, so 24 ÷ 6 = 4 and 24 ÷ 4 = 6
• determine 'factors' for a given whole number, eg the factors of 12 are 1, 2, 3, 4, 6, 12
• use the equals sign to record equivalent number relationships involving multiplication, and to mean 'is the same as', rather than to mean to perform an operation, eg 4 × 3 = 6 × 2
• connect number relationships involving multiplication to factors of a number, eg 'Since 4 × 3 = 6 × 2, then 4, 3, 2 and 6 are factors of 12' (Communicating, Reasoning)
• check number sentences to determine if they are true or false and explain why, eg 'Is 7 × 5 = 8 × 4 true? Why or why not?' (Communicating, Reasoning)
• Develop efficient mental and written strategies, and use appropriate digital technologies, for multiplication and for division where there is no remainder (ACMNA076)
• multiply three or more single-digit numbers, eg 5 × 3 × 6
• model and apply the associative property of multiplication to aid mental computation, eg 2 × 3 × 5 = 2 × 5 × 3 = 10 × 3 = 30
• make generalisations about numbers and number relationships, eg 'It doesn't matter what order you multiply two numbers in because the answer is always the same' (Communicating, Reasoning)
• use mental and informal written strategies to multiply a two-digit number by a one-digit number, including:
• using known facts, eg 10 × 9 = 90, so 13 × 9 = 90 + 9 + 9 + 9 = 90 + 27 = 117
• multiplying the tens and then the units, eg 7 × 19: 7 tens + 7 nines is 70 + 63, which is 133
• using an area model, eg 27 × 8
•   using doubling and repeated doubling to multiply by 2, 4 and 8, eg 23 × 4 is double 23 and then double again
• using the relationship between multiplication facts, eg 41 × 6 is 41 × 3, which is 123, and then double to obtain 246
• factorising the larger number, eg 18 × 5 = 9 × 2 × 5 = 9 × 10 = 90
• create a table or simple spreadsheet to record multiplication facts, eg a 10 × 10 grid showing multiplication facts (Communicating)
• use mental strategies to divide a two-digit number by a one-digit number where there is no remainder, including:
• using the inverse relationship of multiplication and division, eg 63 ÷ 9 = 7 because 7 × 9 = 63
• recalling known division facts
• using halving and repeated halving to divide by 2, 4 and 8, eg 36 ÷ 4: halve 36 and then halve again
• using the relationship between division facts, eg to divide by 5, first divide by 10 and then multiply by 2
• apply the inverse relationship of multiplication and division to justify answers, eg 56 ÷ 8 = 7 because 7 × 8 = 56 (Problem Solving, Reasoning)
• record mental strategies used for multiplication and division
• select and use a variety of mental and informal written strategies to solve multiplication and division problems
• check the answer to a word problem using digital technologies (Reasoning)
• Use mental strategies and informal recording methods for division with remainders
• model division, including where the answer involves a remainder, using concrete materials
• explain why a remainder is obtained in answers to some division problems (Communicating, Reasoning)
• use mental strategies to divide a two-digit number by a one-digit number in problems for which answers include a remainder, eg 27 ÷ 6: if 4 × 6 = 24 and 5 × 6 = 30, the answer is 4 remainder 3
• record remainders to division problems in words, eg 17 ÷ 4 = 4 remainder 1
• interpret the remainder in the context of a word problem, eg 'If a car can safely hold 5 people, how many cars are needed to carry 41 people?'; the answer of 8 remainder 1 means that 9 cars will be needed

### Background Information

An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations.

Linking multiplication and division is an important understanding for students in Stage 2. They should come to realise that division 'undoes' multiplication and multiplication 'undoes' division. Students should be encouraged to check the answer to a division question by multiplying their answer by the divisor. To divide, students may recall division facts or transform the division into a multiplication and use multiplication facts, eg $$35 \div 7$$ is the same as $$\square\; \times\ 7 = 35$$.

The use of digital technologies includes the use of calculators.

### Language

Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, tens, ones, double, multiple, factor, shared between, divide, divided by, division, halve, remainder, equals, is the same as, strategy, digit.

As students become more confident with recalling multiplication facts, they may use less language. For example, 'five rows (or groups) of three' becomes 'five threes' with the 'rows of' or 'groups of' implied. This then leads to 'one three is three', 'two threes are six', 'three threes are nine', and so on.

The term 'product' has a meaning in mathematics that is different from its everyday usage. In mathematics, 'product' refers to the result of multiplying two or more numbers together.

Students need to understand the different uses for the = sign, eg 4 × 3 = 12, where the = sign indicates that the right side of the number sentence contains 'the answer' and should be read to mean 'equals', compared to a statement of equality such as 4 × 3 = 6 × 2, where the = sign should be read to mean 'is the same as'.

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

When working towards the outcome MA2‑6NA the sub-elements (and levels) of Multiplicative strategies (MuS2-MuS7) and Number patterns and algebraic thinking (NPA5-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.