NSW Syllabuses

# Mathematics K–10 - Stage 3 - Number and Algebra Multiplication and Division

## Multiplication and Division 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

• MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

### Content

• Students:
• Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental and written strategies and appropriate digital technologies (ACMNA100)
• use mental and written strategies to multiply three- and four-digit numbers by one-digit 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 three-digit numbers by two-digit numbers, including:
• using an area model for two-digit by two-digit 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 real-life 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 one-digit 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 one-digit 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 one-digit 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 two-digit 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 5-seater 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 two-digit by two-digit 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 sub-elements (and levels) of Multiplicative strategies (MuS5-MuS7) 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:

• 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

• MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

### Content

• 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 two-digit numbers
• select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a one-digit 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?'
• use the term 'operations' to describe collectively the processes of addition, subtraction, multiplication and division
• investigate and establish the order of operations using real-life 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 real-life 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$$\begin{array}{rl}5 + (2 \times 3) &= 5 + 6 \\ &= 11\\\\(2+3) \times (16 - 9) &= 5 \times 7 \\&= 35\\\\3 + \left[ 20 \div (9-5) \right] &= 3 + [20 \div 4 ] \\&= 3 + 5\\&= 8\end{array}$$
• apply the order of operations to perform calculations involving mixed operations and grouping symbols, without the use of digital technologies, eg
$$\begin{array}{rl}32+2-4&=34-4\\&=30\hspace{1.425cm}\text{addition and subtraction only, therefore work from left to right}\\\\32\div2\times4&=16\times4\\&=64\hspace{1.425cm}\text{multiplication and division only, therefore work from left to right}\\\\32\div(2\times4)&=32\div8\\&=4\hspace{1.6cm}\text{perform operation in grouping symbols first}\\\\(32+2)\times4&=34\times4\\&=136\hspace{1.1cm}\text{perform operation in grouping symbols first}\\\\32+2\times4&=32+8\\&=40\hspace{1.425cm}\text{perform multiplication before addition}\end{array}$$
• 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 left-hand 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 sub-elements (and levels) of Multiplicative strategies (MuS6-MuS7) 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.