Multiplication and Division 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

 MA36NA
selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation
 Students:
 Solve problems involving multiplication of large numbers by one or twodigit numbers using efficient mental and written strategies and appropriate digital technologies (ACMNA100)
 use mental and written strategies to multiply three and fourdigit numbers by onedigit numbers, including:

multiplying the thousands, then the hundreds, then the tens and then the ones, eg
\(\begin{align} 673 \times 4&= \left( {600 \times 4} \right) + \left( {70 \times 4} \right) + \left( {3 \times 4} \right)\\&= 2400 + 280 + 12\\&= 2692 \end{align}\) 
using an area model, eg 684 × 5

using the formal algorithm, eg 432 × 5
 use mental and written strategies to multiply two and threedigit numbers by twodigit numbers, including:

using an area model for twodigit by twodigit multiplication, eg 25 × 26
 factorising the numbers, eg 12 × 25 = 3 × 4 × 25 = 3 × 100 = 300

using the extended form (long multiplication) of the formal algorithm, eg
 use digital technologies to multiply numbers of up to four digits
 check answers to mental calculations using digital technologies (Problem Solving)
 apply appropriate mental and written strategies, and digital technologies, to solve multiplication word problems
 use the appropriate operation when solving problems in reallife situations (Problem Solving)
 use inverse operations to justify solutions (Problem Solving, Reasoning)
 record the strategy used to solve multiplication word problems
 use selected words to describe each step of the solution process (Communicating, Problem Solving)
 Solve problems involving division by a onedigit number, including those that result in a remainder (ACMNA101)
 use the term 'quotient' to describe the result of a division calculation, eg 'The quotient when 30 is divided by 6 is 5'
 recognise and use different notations to indicate division, eg \(25÷4\), \(4\overline{\smash{\big)}25}\), \(\frac{25}{4}\)
 record remainders as fractions and decimals, eg \( 25 \div 4 = 6\tfrac{1}{4} \) or 6.25
 use mental and written strategies to divide a number with three or more digits by a onedigit divisor where there is no remainder, including:

dividing the hundreds, then the tens, and then the ones, eg 3248 ÷ 4
\(\begin{align} 3200 \div 4 &= 800\\40 \div 4 &= 10\\8 \div 4 &= 2\\{\text{so }}3248 \div 4 &= 812\end{align}\) 
using the formal algorithm, eg 258 ÷ 6
\(\begin{array}{r}43\\6\overline{\smash{\big)}258} \end{array}\)
 use mental and written strategies to divide a number with three or more digits by a onedigit divisor where there is a remainder, including:

dividing the tens and then the ones, eg 243 ÷ 4
\(\begin{align}240 \div 4 &= 60\\3 \div 4 &= {\textstyle{3 \over 4}}\\{\text{so }}243 \div 4 &= 60{\textstyle{3 \over 4}}\end{align}\) 
using the formal algorithm, eg 587 ÷ 6
\(\begin{align}97&{\textstyle{5 \over 6}}\\6\overline{\smash{\big)}258} \end{align}\)
 explain why the remainder in a division calculation is always less than the number divided by (the divisor) (Communicating, Reasoning)
 show the connection between division and multiplication, including where there is a remainder, eg 25 ÷ 4 = 6 remainder 1, so 25 = 4 × 6 + 1
 use digital technologies to divide whole numbers by one and twodigit divisors
 check answers to mental calculations using digital technologies (Problem Solving)
 apply appropriate mental and written strategies, and digital technologies, to solve division word problems
 recognise when division is required to solve word problems (Problem Solving)
 use inverse operations to justify solutions to problems (Problem Solving, Reasoning)
 use and interpret remainders in solutions to division problems, eg recognise when it is appropriate to round up an answer, such as 'How many 5seater cars are required to take 47 people to the beach?'
 record the strategy used to solve division word problems
 use selected words to describe each step of the solution process (Communicating, Problem Solving)
 Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099)
 round numbers appropriately when obtaining estimates to numerical calculations
 use estimation to check the reasonableness of answers to multiplication and division calculations, eg '32 × 253 will be about, but more than, 30 × 250'
Background Information
Students could extend their recall of number facts beyond the multiplication facts to 10 × 10 by memorising multiples of numbers such as 11, 12, 15, 20 and 25. They could also utilise mental strategies, eg '14 × 6 is 10 sixes plus 4 sixes'.
In Stage 3, mental strategies need to be continually reinforced.
Students may find recording (writing out) informal mental strategies to be more efficient than using formal written algorithms, particularly in the case of multiplication.
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 area model for twodigit by twodigit multiplication in Stage 3 is a precursor to the use of the area model for the expansion of binomial products in Stage 5.
Language
Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, area, thousands, hundreds, tens, ones, double, multiple, factor, divide, divided by, quotient, division, halve, remainder, fraction, decimal, equals, strategy, digit, estimate, round to.
In mathematics, 'quotient' refers to the result of dividing one number by another.
Teachers should model and use a variety of expressions for multiplication and division. They should draw students' attention to the fact that the words used for division may require the operation to be performed with the numbers in the reverse order to that in which they are stated in the question. For example, 'divide 6 by 2' and '6 divided by 2' require the operation to be performed with the numbers in the same order as they are presented in the question (ie 6 ÷ 2). However, 'How many 2s in 6?' requires the operation to be performed with the numbers in the reverse order to that in which they are stated in the question (ie 6 ÷ 2).
The terms 'ratio' and 'rate' are not introduced until Stage 4, but students need to be able to interpret problems involving simple rates as requiring multiplication or division.
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA3‑6NA the subelements (and levels) of Multiplicative strategies (MuS5MuS7) 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.
Multiplication and Division 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

 MA36NA
selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation
 Students:
 Select and apply efficient mental and written strategies, and appropriate digital technologies, to solve problems involving multiplication and division with whole numbers (ACMNA123)
 select and use efficient mental and written strategies, and digital technologies, to multiply whole numbers of up to four digits by one and twodigit numbers
 select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a onedigit divisor, including where there is a remainder
 estimate solutions to problems and check to justify solutions (Problem Solving, Reasoning)
 use mental strategies to multiply and divide numbers by 10, 100, 1000 and their multiples
 solve word problems involving multiplication and division, eg 'A recipe requires 3 cups of flour for 10 people. How many cups of flour are required for 40 people?'
 use appropriate language to compare quantities, eg 'twice as much as', 'half as much as' (Communicating)
 use a table or similar organiser to record methods used to solve problems (Communicating, Problem Solving)
 recognise symbols used to record speed in kilometres per hour, eg 80 km/h
 solve simple problems involving speed, eg 'How long would it take to travel 600 km if the average speed for the trip is 75 km/h?'
 Explore the use of brackets and the order of operations to write number sentences (ACMNA134)
 use the term 'operations' to describe collectively the processes of addition, subtraction, multiplication and division
 investigate and establish the order of operations using reallife contexts, eg 'I buy six goldfish costing $10 each and two water plants costing $4 each. What is the total cost?'; this can be represented by the number sentence 6 × 10 + 2 × 4 but, to obtain the total cost, multiplication must be performed before addition
 write number sentences to represent reallife situations (Communicating, Problem Solving)
 recognise that the grouping symbols \( \big( \big) \) and \( \big[ \; \big] \) are used in number sentences to indicate operations that must be performed first
 recognise that if more than one pair of grouping symbols are used, the operation within the innermost grouping symbols is performed first
 perform calculations involving grouping symbols without the use of digital technologies, eg\( \begin{array}{rl}5 + (2 \times 3) &= 5 + 6 \\ &= 11\\\\(2+3) \times (16  9) &= 5 \times 7 \\&= 35\\\\3 + \left[ 20 \div (95) \right] &= 3 + [20 \div 4 ] \\&= 3 + 5\\&= 8\end{array}\)

apply the order of operations to perform calculations involving mixed operations and grouping symbols, without the use of digital technologies, eg
\(\begin{array}{rl}32+24&=344\\&=30\hspace{1.425cm}\text{addition and subtraction only, therefore work from left to right}\\\\32\div2\times4&=16\times4\\&=64\hspace{1.425cm}\text{multiplication and division only, therefore work from left to right}\\\\32\div(2\times4)&=32\div8\\&=4\hspace{1.6cm}\text{perform operation in grouping symbols first}\\\\(32+2)\times4&=34\times4\\&=136\hspace{1.1cm}\text{perform operation in grouping symbols first}\\\\32+2\times4&=32+8\\&=40\hspace{1.425cm}\text{perform multiplication before addition}\end{array}\)
 investigate whether different digital technologies apply the order of operations (Reasoning)
 recognise when grouping symbols are not necessary, eg 32 + (2 × 4) has the same answer as 32 + 2 × 4
Background Information
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, or by utilising mental strategies, eg '14 × 6 is 10 sixes plus 4 sixes'.
The simplest multiplication word problems relate to rates, eg 'If four students earn $3 each, how much do they have all together?' Another type of problem is related to ratio and uses language such as 'twice as many as' and 'six times as many as'.
An 'operation' is a mathematical process. The four basic operations are addition, subtraction, multiplication and division. Other operations include raising a number to a power and taking a root of a number. An 'operator' is a symbol that indicates the type of operation, eg +, –, × and ÷.
Refer also to background information in Multiplication and Division 1.
Language
Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, area, thousands, hundreds, tens, ones, double, multiple, factor, divide, divided by, quotient, division, halve, remainder, fraction, decimal, equals, strategy, digit, estimate, speed, per, operations, order of operations, grouping symbols, brackets, number sentence, is the same as.
When solving word problems, students should be encouraged to write a few key words on the lefthand side of the equals sign to identify what is being found in each step of their working, eg 'cost of goldfish = …', 'cost of plants = …', 'total cost = …'.
'Grouping symbols' is a collective term used to describe brackets \( \big[ \; \big] \), parentheses \( \big( \; \big) \) and braces \( \big\{ \; \big\} \). The term 'brackets' is often used in place of 'parentheses'.
Often in mathematics when grouping symbols have one level of nesting, the inner pair is parentheses \( \big( \; \big) \) and the outer pair is brackets\( \big[ \; \big] \), eg \(360 \div \left[ {4 \times \left( {20  11} \right)}\right]\).
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA3‑6NA the subelements (and levels) of Multiplicative strategies (MuS6MuS7) 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.